Content-Type: multipart/mixed; boundary="-------------0409230749612" This is a multi-part message in MIME format. ---------------0409230749612 Content-Type: text/plain; name="04-302.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-302.keywords" multisummability, power series, linear partial differential equations, Cauchy problem, analytic continuation ---------------0409230749612 Content-Type: application/postscript; name="pdesum1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pdesum1.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: pdesum1.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMSY10 CMR12 CMBX10 CMR10 CMBX12 CMMI10 CMR8 %%+ CMMI8 CMTI10 CMR6 CMSY6 CMR9 CMEX10 CMMI6 MSBM10 CMSY8 LASY10 %%+ TeX-cmex8 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o pdesum1.ps pdesum1.dvi %DVIPSParameters: dpi=600, 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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5593 y Fk(1)p Fg(=q)926 5601 y(p)836 5644 y(p)1008 5587 y Fq(in)37 b(the)g(set)g Fe(C)1498 5554 y Fn(p)1562 5587 y Fl(n)25 b Fq(\002,)38 b(where)f(\002)f(is)h (the)g(singular)g(set)g(of)g(the)g(analytic)i(function)75 5740 y Fp(\025)p Fq(\()p Fp(u)215 5696 y Fn(q)247 5705 y Fk(1)215 5766 y Fo(1)286 5740 y Fp(;)15 b(:)g(:)g(:)h(;)f(u)539 5691 y Fn(q)571 5699 y Fg(p)539 5752 y Fn(p)612 5740 y Fq(\),)30 b(for)h Fp(R)911 5755 y Fn(k)979 5740 y Fl(\025)25 b Fp(R)1144 5754 y Fo(0)1213 5740 y Fq(and)30 b Fp(R)1460 5707 y Fd(0)1483 5755 y Fn(k)1551 5740 y Fl(\025)25 b Fp(R)1716 5754 y Fo(0)1755 5740 y Fq(,)31 b(1)26 b Fl(\024)f Fp(k)j Fl(\024)d Fp(p)30 b Fq(.)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 75 100 a Fq(4)216 399 y(Due)39 b(to)h(the)f(assumption)f Ff(A)h Fq(ii\),)j(there)d(exist)g(a)h (constan)m(t)g Fp(k)2430 413 y Fn(j)2506 399 y Fp(>)f Fq(0,)i(suc)m(h)e(that)g(for)g Fp(n)3349 414 y Fn(k)3430 399 y Fq(su\016cien)m(tly)75 511 y(large,)32 b(1)25 b Fl(\024)g Fp(k)k Fl(\024)c Fp(p)30 b Fq(and)g(for)g Fp(R)1122 526 y Fn(k)1190 511 y Fq(=)25 b Fp(R)1355 525 y Fn(n)1398 537 y Fg(k)1465 511 y Fq(=)g Fp(n)1616 526 y Fn(k)1658 511 y Fp(=)p Fl(j)p Fp(z)1770 526 y Fn(k)1814 511 y Fl(j)p Fq(,)31 b(1)25 b Fl(\024)g Fp(k)k Fl(\024)c Fp(p)p Fq(,)592 781 y Fl(j)p Fq(\005)685 737 y Fn(p)685 810 y(k)r Fo(=1)818 781 y Fp(n)873 796 y Fn(k)915 781 y Fq(!)p Fp(A)1008 795 y Fn(j;n)1100 804 y Fk(1)1134 795 y Fn(;:::)n(;n)1275 803 y Fg(p)1315 781 y Fl(j)h(\024)f Fp(k)1509 795 y Fn(j)1545 781 y Fq(\005)1613 737 y Fn(p)1613 810 y(k)r Fo(=1)1746 781 y Fq(\()p Fp(n)1836 796 y Fn(k)1879 781 y Fp(=)p Fl(j)p Fp(z)1991 796 y Fn(k)2035 781 y Fl(j)p Fq(\))2095 743 y Fn(m)2157 755 y Fg(k)2196 743 y Fn(j)2232 781 y Fq(\005)2300 737 y Fn(p)2300 810 y(k)r Fo(=1)2433 781 y Fl(j)p Fp(z)2500 796 y Fn(k)2544 781 y Fl(j)2569 739 y Fn(n)2612 751 y Fg(k)2654 781 y Fq(\005)2722 737 y Fn(p)2722 810 y(k)r Fo(=1)2855 781 y Fp(n)2910 796 y Fn(k)2952 781 y Fq(!)2993 716 y(\005)3061 672 y Fn(p)3061 745 y(k)r Fo(=1)3194 716 y Fp(e)3236 683 y Fn(n)3279 695 y Fg(k)p 2987 760 341 4 v 2987 844 a Fq(\005)3055 799 y Fn(p)3055 873 y(k)r Fo(=1)3188 844 y Fp(n)3243 801 y Fn(n)3286 813 y Fg(k)3243 873 y Fn(k)75 1064 y Fq(Due)39 b(to)g(the)g(Stirling)g(form)m(ula)g Fp(n)p Fq(!)g Fl(\030)g Fp(n)1516 1031 y Fn(n)1563 1064 y Fp(e)1605 1031 y Fd(\000)p Fn(n)1707 1064 y Fq(\(2)p Fp(\031)s(n)p Fq(\))1932 1031 y Fo(1)p Fn(=)p Fo(2)2081 1064 y Fq(for)g Fp(n)f Fl(!)i(1)p Fq(,)g(There)f(exists)g(constan)m(ts)3553 1040 y(~)3551 1064 y Fp(k)3598 1078 y Fn(j)3674 1064 y Fp(>)g Fq(0,)92 1177 y(~)-62 b Fp(m)155 1192 y Fn(k)r(j)255 1177 y Fp(>)25 b Fq(0,)31 b(1)26 b Fl(\024)f Fp(k)j Fl(\024)d Fp(p)30 b Fq(suc)m(h)g(that)h(for)f Fp(n)1462 1192 y Fn(k)1535 1177 y Fq(su\016cien)m(tly)h(large,)h(1)25 b Fl(\024)g Fp(k)k Fl(\024)c Fp(p)p Fq(,)75 1388 y(\()p Fp(n)p Fq(!)p Fp(A)258 1402 y Fn(j)295 1388 y Fq(\))593 b Fl(j)p Fq(\005)1016 1344 y Fn(p)1016 1417 y(k)r Fo(=1)1149 1388 y Fp(n)1204 1403 y Fn(k)1246 1388 y Fq(!)p Fp(A)1339 1402 y Fn(j;n)1431 1411 y Fk(1)1465 1402 y Fn(;:::)o(;n)1607 1410 y Fg(p)1647 1388 y Fl(j)25 b(\024)1795 1364 y Fq(~)1793 1388 y Fp(k)1840 1402 y Fn(j)1877 1388 y Fq(\005)1945 1344 y Fn(p)1945 1417 y(k)r Fo(=1)2078 1388 y Fq(\()p Fp(n)2168 1403 y Fn(k)2210 1388 y Fp(=)p Fl(j)p Fp(z)2322 1403 y Fn(k)2366 1388 y Fl(j)p Fq(\))2440 1350 y Fo(~)-49 b Fn(m)2488 1362 y Fg(k)q(j)2560 1388 y Fq(\005)2628 1344 y Fn(p)2628 1417 y(k)r Fo(=1)2761 1388 y Fl(j)p Fp(z)2828 1403 y Fn(k)2871 1388 y Fl(j)2896 1346 y Fn(n)2939 1358 y Fg(k)2981 1388 y Fp(:)75 1599 y Fq(On)33 b(the)g(other)h(hand,)f (due)g(to)h(the)g(m)m(ultidimensional)g(Cauc)m(h)m(y)g(form)m(ula)g (\(see)g([5]\),)i(there)d(exists)h Fp(M)41 b(>)30 b Fq(0)75 1712 y(and)g Fp(c)25 b(>)g Fq(0)31 b(suc)m(h)f(that,)h(for)f(all)i Fp(n)1236 1727 y Fn(k)1278 1712 y Fq(,)f(1)25 b Fl(\024)g Fp(k)k Fl(\024)c Fp(p)p Fq(,)75 1923 y(\(\010\))1281 b Fl(j)p Fq(\010)1583 1937 y Fn(n)1626 1946 y Fk(1)1661 1937 y Fn(;:::)n(;n)1802 1945 y Fg(p)1842 1923 y Fl(j)25 b(\024)g Fp(M)10 b Fq(\005)2154 1879 y Fn(p)2154 1952 y(k)r Fo(=1)2287 1923 y Fp(c)2326 1885 y Fn(n)2369 1897 y Fg(k)2412 1923 y Fp(:)75 2134 y Fq(F)-8 b(rom)34 b(the)g(estimates)h (\()p Fp(n)p Fq(!)p Fp(A)1063 2148 y Fn(j)1100 2134 y Fq(\))f(and)f(\(\010\),)i(w)m(e)f(deduce)f(that)h(\()p Fp(u)2275 2148 y Fn(j)2312 2134 y Fq(\))g(con)m(v)m(erges)h(absolutely) g(and)e(lo)s(cally)i(uni-)75 2247 y(formly)30 b(in)g(\()p Fp(z)546 2261 y Fo(1)586 2247 y Fp(;)15 b(:)g(:)g(:)i(;)e(z)830 2261 y Fn(p)870 2247 y Fq(\))26 b Fl(2)f Fp(D)1092 2261 y Fn(\032)1128 2270 y Fk(1)1187 2247 y Fl(\002)20 b(\001)15 b(\001)g(\001)21 b(\002)f Fp(D)1570 2261 y Fn(\032)1606 2269 y Fg(p)1647 2247 y Fq(.)216 2416 y(F)-8 b(or)21 b Fp(j)31 b Fl(\025)25 b Fp(\024)p Fq(,)e(consider)e(the)f(sequence)h Fp(\025)1544 2383 y Fn(j)1581 2416 y Fq(\()p Fp(w)1681 2430 y Fo(1)1721 2416 y Fp(;)15 b(:)g(:)g(:)h(;)f(w)1987 2430 y Fn(p)2027 2416 y Fq(\).)38 b(Due)21 b(to)g(the)g(assumption)f Ff(A)g Fq(iii\),)k Fp(\025)3336 2383 y Fn(j)3373 2416 y Fq(\()p Fp(w)3473 2430 y Fo(1)3512 2416 y Fp(;)15 b(:)g(:)g(:)i(;)e (w)3779 2430 y Fn(p)3819 2416 y Fq(\))75 2529 y(satis\014es)75 2787 y(\()p Fp(\025)163 2754 y Fn(j)200 2787 y Fq(\))415 b Fp(p)p Fq(\()p Fp(w)796 2801 y Fo(1)836 2787 y Fp(;)15 b(:)g(:)g(:)i(;)e(w)1103 2801 y Fn(p)1143 2787 y Fq(\))p Fp(\025)1231 2749 y Fn(j)1268 2787 y Fq(\()p Fp(w)1368 2801 y Fo(1)1408 2787 y Fp(;)g(:)g(:)g(:)h(;)f(w)1674 2801 y Fn(p)1714 2787 y Fq(\))26 b(=)1916 2673 y Fn(\024)1871 2701 y Fh(X)1872 2895 y Fn(\027)t Fo(=1)2017 2787 y Fp(p)2063 2801 y Fn(\027)2106 2787 y Fq(\()p Fp(w)2206 2801 y Fo(1)2246 2787 y Fp(;)15 b(:)g(:)g(:)i(;)e(w)2513 2801 y Fn(p)2553 2787 y Fq(\))p Fp(\025)2641 2749 y Fn(j)t Fd(\000)p Fn(\027)2772 2787 y Fq(\()p Fp(w)2872 2801 y Fo(1)2912 2787 y Fp(;)g(:)g(:)g(:)h(;)f (w)3178 2801 y Fn(p)3218 2787 y Fq(\))p Fp(:)75 3079 y Fq(F)-8 b(rom)28 b(this)f(latter)h(equation,)h(w)m(e)e(deduce)g(that) h Fp(u)1799 3093 y Fn(j)1835 3079 y Fq(\()p Fp(z)1912 3093 y Fo(1)1953 3079 y Fp(;)15 b(:)g(:)g(:)h(;)f(z)2196 3093 y Fn(p)2237 3079 y Fq(\))27 b(satisfy)g(the)h(relation)g(\(3\))g (\()g(the)f(deriv)-5 b(ations)75 3192 y(under)33 b(the)i(signs)726 3123 y Fh(P)857 3192 y Fq(and)1038 3118 y Fh(H)1133 3192 y Fq(are)h(allo)m(w)m(ed)g(b)s(ecause)f(of)g(uniformly)f(con)m(v)m (ergence)j(of)e(the)h(series\).)54 b(Hence,)75 3304 y(the)31 b(formal)f(series)1249 3471 y(^)-51 b Fp(u)p Fq(\()p Fp(t;)15 b(z)1445 3485 y Fo(1)1485 3471 y Fp(;)g(:)g(:)g(:)i(;)e(z)1729 3485 y Fn(p)1769 3471 y Fq(\))26 b(=)1956 3357 y Fd(1)1926 3385 y Fh(X)1930 3580 y Fn(j)t Fo(=0)2082 3409 y Fp(t)2115 3376 y Fn(j)p 2082 3450 70 4 v 2083 3533 a Fp(j)5 b Fq(!)2162 3471 y Fp(u)2214 3485 y Fn(j)2250 3471 y Fq(\()p Fp(z)2327 3485 y Fo(1)2367 3471 y Fp(;)15 b(:)g(:)g(:)i(;)e(z)2611 3485 y Fn(p)2651 3471 y Fq(\))75 3740 y(is)30 b(a)h(formal)g(solution)g (of)f(\(1\).)42 b(Due)31 b(to)g(the)f(residues)g(theorem,)h(w)m(e)g(ha) m(v)m(e)h(for)e(1)25 b Fl(\024)g Fp(k)k Fl(\024)c Fp(p)p Fq(,)1409 3933 y Fp(n)1464 3948 y Fn(k)1506 3933 y Fq(!)p 1328 3973 286 4 v 1328 4056 a(\(2)p Fp(\031)s(i)p Fq(\))p Fp(q)1570 4071 y Fn(k)1638 3870 y Fh(I)1689 4077 y Fn(q)1721 4089 y Fg(k)1759 4077 y Fn(C)1809 4088 y Fg(R)1855 4105 y(k)1945 3933 y Fp(e)1987 3900 y Fn(w)2038 3912 y Fg(k)2076 3900 y Fn(z)2109 3912 y Fg(k)p 1926 3973 243 4 v 1926 4070 a Fp(w)1993 4027 y Fn(n)2036 4039 y Fg(k)2075 4027 y Fo(+1)1991 4099 y Fn(k)2179 3994 y Fp(dw)2291 4009 y Fn(k)2359 3994 y Fq(=)g Fp(z)2501 3951 y Fn(n)2544 3963 y Fg(k)2497 4023 y Fn(k)2587 3994 y Fp(;)75 4280 y Fq(from)30 b(whic)m(h)g(w)m(e)h(deduce)f(that)37 b(^)-51 b Fp(u)p Fq(\(0)p Fp(;)15 b(z)1398 4294 y Fo(1)1439 4280 y Fp(;)g(:)g(:)g(:)h(;)f(z)1682 4294 y Fn(p)1723 4280 y Fq(\))25 b(=)g(\010\()p Fp(z)2022 4294 y Fo(1)2062 4280 y Fp(;)15 b(:)g(:)g(:)i(;)e(z)2306 4294 y Fn(p)2346 4280 y Fq(\).)1350 b Fc(2)75 4633 y Fr(2)135 b(Gevrey)45 b(estimates)75 4841 y Fq(In)39 b(the)i(follo)m(wing)g(theorem,)j(w)m(e) c(giv)m(e)i(Gevrey)f(estimates)g(for)f(the)h(co)s(e\016cien)m(ts)h(of)e (the)g(formal)h(p)s(o)m(w)m(ers)75 4954 y(series)31 b(\(2\))g (solutions)g(of)f(the)h(giv)m(en)g(partial)h(di\013eren)m(tial)f (equation)h(\(1\).)75 5180 y Ff(Theorem)j(1)46 b Fm(Under)35 b(the)g(assumption)i Ff(A)p Fm(,)d(for)i(al)5 b(l)35 b Fp(\032)1999 5195 y Fn(k)2071 5180 y Fp(>)29 b Fq(0)35 b Fm(su\016ciently)g(smal)5 b(l,)36 b Fq(1)30 b Fl(\024)e Fp(k)33 b Fl(\024)c Fp(p)p Fm(,)35 b(ther)-5 b(e)35 b(exist)75 5293 y Fp(C)q(;)15 b(K)33 b(>)25 b Fq(0)33 b Fm(such)f(that)1204 5482 y Fl(j)p Fp(u)1281 5496 y Fn(j)1318 5482 y Fq(\()p Fp(z)1395 5496 y Fo(1)1435 5482 y Fp(;)15 b(:)g(:)g(:)h(;)f(z)1678 5496 y Fn(p)1719 5482 y Fq(\))p Fl(j)26 b(\024)f Fp(C)7 b(K)2057 5444 y Fn(j)2154 5363 y(p)2114 5395 y Fh(Y)2107 5593 y Fn(k)r Fo(=1)2251 5482 y Fq(\000\(1)21 b(+)f Fp(m)2580 5497 y Fn(k)2622 5482 y Fp(j)5 b Fq(\))p Fp(;)75 5740 y Fm(for)33 b(al)5 b(l)33 b Fp(j)e Fl(\025)25 b Fq(0)p Fm(,)33 b Fl(j)p Fp(z)687 5755 y Fn(k)730 5740 y Fl(j)26 b(\024)f Fp(\032)924 5755 y Fn(k)966 5740 y Fm(,)33 b Fq(1)25 b Fl(\024)g Fp(k)k Fl(\024)c Fp(p)p Fm(.)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 3809 100 a Fq(5)75 399 y Ff(Pro)s(of)31 b Fq(Due)g(to)g(the)g(assumption)e Ff(A)i Fq(ii\),)g(there)g(exists)g Fp(C)g(>)25 b Fq(0)31 b(suc)m(h)f(that)1273 593 y Fl(j)p Fp(\025)1351 555 y Fn(j)1388 593 y Fq(\()p Fp(w)1488 607 y Fo(1)1528 593 y Fp(;)15 b(:)g(:)g(:)h(;)f(w)1794 607 y Fn(p)1835 593 y Fq(\))p Fl(j)26 b(\024)e Fp(C)2088 555 y Fn(pj)2160 593 y Fq(\005)2228 549 y Fn(p)2228 622 y(k)r Fo(=1)2361 593 y Fl(j)p Fp(w)2451 608 y Fn(k)2494 593 y Fl(j)2519 555 y Fn(m)2581 567 y Fg(k)2620 555 y Fn(j)75 797 y Fq(for)30 b Fl(j)p Fp(w)304 812 y Fn(k)347 797 y Fl(j)c(\025)f Fp(R)563 811 y Fo(0)602 797 y Fq(,)31 b(1)25 b Fl(\024)g Fp(k)k Fl(\024)c Fp(p)p Fq(.)40 b(There)30 b(exists)1623 773 y(~)1621 797 y Fp(k)j Fq(suc)m(h)e(that)f(for)h Fl(j)p Fp(z)2310 812 y Fn(k)2353 797 y Fl(j)25 b(\024)g Fp(\032)2546 812 y Fn(k)2619 797 y Fq(and)30 b Fp(R)2865 811 y Fn(n)2908 823 y Fg(k)2975 797 y Fq(=)25 b Fp(n)3126 812 y Fn(k)3169 797 y Fp(=\032)3261 812 y Fn(k)3304 797 y Fq(,)30 b(1)c Fl(\024)f Fp(k)j Fl(\024)d Fp(p)p Fq(,)596 1060 y Fl(j)732 995 y Fq(\005)800 951 y Fn(p)800 1024 y(k)r Fo(=1)933 995 y Fp(n)988 1010 y Fn(k)1031 995 y Fq(!)p 631 1039 527 4 v 631 1123 a(\(2)p Fp(\031)s(i)p Fq(\))832 1096 y Fn(p)873 1123 y Fq(\005)941 1078 y Fn(p)941 1152 y(k)r Fo(=1)1074 1123 y Fp(q)1115 1138 y Fn(k)1167 1060 y Fp(A)1235 1074 y Fn(j;n)1327 1083 y Fk(1)1361 1074 y Fn(;:::)n(;n)1502 1082 y Fg(p)1542 1060 y Fl(j)h(\024)1784 995 y Fq(\005)1852 951 y Fn(p)1852 1024 y(k)r Fo(=1)1985 995 y Fp(n)2040 1010 y Fn(k)2083 995 y Fq(!)p 1699 1039 496 4 v 1699 1123 a(\(2)p Fp(\031)s Fq(\))1869 1096 y Fn(p)1909 1123 y Fq(\005)1977 1078 y Fn(p)1977 1152 y(k)r Fo(=1)2110 1123 y Fp(q)2151 1138 y Fn(k)2206 1036 y Fq(~)2204 1060 y Fp(k)s(C)2326 1022 y Fn(pj)2407 995 y Fq(\005)2475 951 y Fn(p)2475 1024 y(k)r Fo(=1)2608 995 y Fq(\()p Fp(R)2712 1009 y Fn(n)2755 1021 y Fg(k)2798 995 y Fq(\))2833 962 y Fn(m)2895 974 y Fg(k)2934 962 y Fn(j)p 2407 1039 563 4 v 2511 1123 a Fq(\005)2579 1078 y Fn(p)2579 1152 y(k)r Fo(=1)2712 1123 y Fp(R)2782 1080 y Fn(n)2825 1092 y Fg(k)2781 1134 y Fn(n)2824 1146 y Fg(k)2980 1060 y Fq(\005)3048 1015 y Fn(p)3048 1089 y(k)r Fo(=1)3181 1060 y Fp(e)3223 1022 y Fn(n)3266 1034 y Fg(k)3309 1060 y Fp(:)75 1272 y Fq(Due)31 b(to)g(the)f(F)-8 b(ubini)30 b(theorem,)h(for)g(all)g Fp(x)25 b(>)g Fq(0,)1299 1418 y Fd(1)1269 1445 y Fh(X)1273 1640 y Fn(j)t Fo(=0)1688 1470 y Fp(x)1740 1437 y Fn(pj)p 1425 1510 651 4 v 1425 1594 a Fq(\005)1493 1550 y Fn(p)1493 1623 y(k)r Fo(=1)1626 1594 y Fq(\000\(1)c(+)f Fp(m)1955 1609 y Fn(k)1997 1594 y Fp(j)5 b Fq(\))2085 1531 y Fl(j)p Fp(u)2162 1545 y Fn(j)2199 1531 y Fq(\()p Fp(z)2276 1545 y Fo(1)2316 1531 y Fp(;)15 b(:)g(:)g(:)i(;)e(z)2560 1545 y Fn(p)2600 1531 y Fq(\))p Fl(j)706 1814 y(\024)804 1790 y Fq(~)802 1814 y Fp(k)1004 1700 y Fd(1)973 1727 y Fh(X)867 1922 y Fn(n)910 1931 y Fk(1)944 1922 y Fn(;:::)o(;n)1086 1930 y Fg(p)1121 1922 y Fo(=0)1237 1746 y Fl(j)p Fq(\010)1328 1760 y Fn(n)1371 1769 y Fk(1)1405 1760 y Fn(;:::)n(;n)1546 1768 y Fg(p)1586 1746 y Fl(j)p Fq(\005)1679 1702 y Fn(p)1679 1776 y(k)r Fo(=1)1812 1746 y Fp(n)1867 1761 y Fn(k)1909 1746 y Fq(!)p Fp(e)1976 1714 y Fn(n)2019 1726 y Fg(k)p 1237 1793 826 4 v 1471 1876 a Fq(\005)1539 1832 y Fn(p)1539 1906 y(k)r Fo(=1)1672 1876 y Fp(R)1742 1833 y Fn(n)1785 1845 y Fg(k)1741 1888 y Fn(n)1784 1900 y Fg(k)2092 1814 y Fl(\002)2214 1700 y Fd(1)2183 1727 y Fh(X)2188 1923 y Fn(j)t Fo(=0)2315 1814 y Fq(\(\005)2418 1769 y Fn(p)2418 1843 y(k)r Fo(=1)2561 1752 y Fq(\()p Fp(xC)7 b Fq(\))2755 1719 y Fn(j)2792 1752 y Fq(\()p Fp(R)2896 1766 y Fn(n)2939 1778 y Fg(k)2981 1752 y Fq(\))3016 1719 y Fn(m)3078 1731 y Fg(k)3117 1719 y Fn(j)p 2561 1793 593 4 v 2633 1876 a Fq(\000\(1)20 b(+)g Fp(m)2961 1891 y Fn(k)3004 1876 y Fp(j)5 b Fq(\))3163 1814 y(\))p Fp(:)216 2123 y Fq(Due)24 b(to)g(the)f(estimates)i(of)e(the)h(Mittag)h(Le\017er)f(functions)e (\(see)j([3])f(p.)38 b(233\),)26 b(there)e(exist)3308 2099 y(~)3306 2123 y Fp(k)3353 2138 y Fn(k)3396 2123 y Fq(,)g(1)i Fl(\024)f Fp(k)k Fl(\024)c Fp(p)p Fq(,)75 2236 y(suc)m(h)30 b(that)946 2287 y Fd(1)916 2314 y Fh(X)920 2510 y Fn(j)t Fo(=0)1047 2400 y Fq(\()1129 2282 y Fn(p)1088 2314 y Fh(Y)1082 2512 y Fn(k)r Fo(=1)1236 2339 y Fq(\()p Fp(xC)7 b Fq(\))1430 2306 y Fn(j)1467 2339 y Fq(\()p Fp(R)1571 2353 y Fn(n)1614 2365 y Fg(k)1656 2339 y Fq(\))1691 2306 y Fn(m)1753 2318 y Fg(k)1792 2306 y Fn(j)p 1236 2380 V 1308 2463 a Fq(\000\(1)21 b(+)f Fp(m)1637 2478 y Fn(k)1679 2463 y Fp(j)5 b Fq(\))1839 2400 y(\))83 b Fl(\024)2157 2282 y Fn(p)2117 2314 y Fh(Y)2111 2512 y Fn(k)r Fo(=1)2285 2287 y Fd(1)2255 2314 y Fh(X)2259 2510 y Fn(j)t Fo(=0)2411 2339 y Fq(\()p Fp(xC)7 b Fq(\))2605 2306 y Fn(j)2642 2339 y Fq(\()p Fp(R)2746 2353 y Fn(n)2789 2365 y Fg(k)2831 2339 y Fq(\))2866 2306 y Fn(m)2928 2318 y Fg(k)2967 2306 y Fn(j)p 2411 2380 V 2483 2463 a Fq(\000\(1)21 b(+)f Fp(m)2812 2478 y Fn(k)2854 2463 y Fp(j)5 b Fq(\))1957 2688 y Fl(\024)2219 2569 y Fn(p)2179 2601 y Fh(Y)2173 2799 y Fn(k)r Fo(=1)2301 2688 y Fq(\()2338 2664 y(~)2336 2688 y Fp(k)2383 2703 y Fn(k)2427 2688 y Fp(e)2469 2650 y Fo(\()p Fn(xc)p Fo(\))2594 2626 y Fk(1)p Fg(=m)2710 2643 y(k)2753 2650 y Fn(R)2806 2658 y Fg(n)2844 2676 y(k)2891 2688 y Fq(\))p Fp(:)75 2929 y Fq(W)-8 b(e)32 b(deduce)e(that)h(there)f(exists)h Fp(K)h(>)25 b Fq(0)31 b(suc)m(h)f(that)595 3067 y Fd(1)564 3095 y Fh(X)569 3290 y Fn(j)t Fo(=0)984 3119 y Fp(x)1036 3086 y Fn(pj)p 721 3160 651 4 v 721 3244 a Fq(\005)789 3200 y Fn(p)789 3273 y(k)r Fo(=1)922 3244 y Fq(\000\(1)21 b(+)e Fp(m)1250 3259 y Fn(k)1293 3244 y Fp(j)5 b Fq(\))1381 3181 y Fl(j)p Fp(u)1458 3195 y Fn(j)1495 3181 y Fq(\()p Fp(z)1572 3195 y Fo(1)1612 3181 y Fp(;)15 b(:)g(:)g(:)h(;)f(z)1855 3195 y Fn(p)1896 3181 y Fq(\))p Fl(j)26 b(\024)f Fp(K)2223 3062 y Fn(p)2183 3095 y Fh(Y)2176 3292 y Fn(k)r Fo(=1)2371 3067 y Fd(1)2340 3095 y Fh(X)2320 3289 y Fn(n)2363 3301 y Fg(k)2401 3289 y Fo(=0)2517 3119 y Fp(c)2556 3086 y Fn(n)2599 3098 y Fg(k)2641 3119 y Fp(n)2696 3134 y Fn(k)2738 3119 y Fq(!)p Fp(e)2805 3086 y Fn(n)2848 3098 y Fg(k)2891 3119 y Fp(e)2933 3084 y Fo(\()p Fn(xc)p Fo(\))3058 3059 y Fk(1)p Fg(=m)3174 3077 y(k)3217 3084 y Fn(R)3270 3092 y Fg(n)3308 3110 y(k)p 2517 3160 839 4 v 2858 3243 a Fp(R)2928 3200 y Fn(n)2971 3212 y Fg(k)2927 3255 y Fn(n)2970 3267 y Fg(k)75 3465 y Fq(Using)g(the)h(Stirling)f(form)m(ula,)i(one)f (can)f(sho)m(w)g(that)h(the)g(righ)m(t)g(handside)e(of)h(the)h(latter)g (estimate)i(con)m(v)m(erges)75 3578 y(for)i Fl(j)p Fp(x)p Fl(j)h Fq(su\016cien)m(tly)g(small.)41 b(Hence,)32 b(the)e(function) 1141 3842 y Fp( )s Fq(\()p Fp(x)p Fq(\))c(=)1478 3728 y Fd(1)1447 3755 y Fh(X)1452 3951 y Fn(j)t Fo(=0)1867 3780 y Fp(x)1919 3747 y Fn(pj)p 1604 3821 651 4 v 1604 3905 a Fq(\005)1672 3860 y Fn(p)1672 3934 y(k)r Fo(=1)1805 3905 y Fq(\000\(1)20 b(+)g Fp(m)2133 3920 y Fn(k)2176 3905 y Fp(j)5 b Fq(\))2264 3842 y Fp(u)2316 3856 y Fn(j)2352 3842 y Fq(\()p Fp(z)2429 3856 y Fo(1)2469 3842 y Fp(;)15 b(:)g(:)g(:)i(;)e(z)2713 3856 y Fn(p)2753 3842 y Fq(\))75 4131 y(is)38 b(holomorphic)f(for)g Fp(x)h Fq(suc)m(h)f(that)h Fl(j)p Fp(x)p Fl(j)g Fq(is)g(small.)62 b(F)-8 b(rom)39 b(the)e(Cauc)m(h)m(y)h(form)m(ula,)i(w)m(e)e(deduce)f(that)h(there)75 4244 y(exist)31 b Fp(C)q(;)15 b(K)33 b(>)25 b Fq(0)31 b(suc)m(h)f(that)1204 4507 y Fl(j)p Fp(u)1281 4521 y Fn(j)1318 4507 y Fq(\()p Fp(z)1395 4521 y Fo(1)1435 4507 y Fp(;)15 b(:)g(:)g(:)h(;)f(z)1678 4521 y Fn(p)1719 4507 y Fq(\))p Fl(j)26 b(\024)f Fp(C)7 b(K)2057 4470 y Fn(j)2154 4389 y(p)2114 4421 y Fh(Y)2107 4619 y Fn(k)r Fo(=1)2251 4507 y Fq(\000\(1)21 b(+)f Fp(m)2580 4522 y Fn(k)2622 4507 y Fp(j)5 b Fq(\))p Fp(;)75 4786 y Fq(for)30 b(all)h Fp(j)g Fl(\025)25 b Fq(0,)31 b Fl(j)p Fp(z)672 4801 y Fn(k)715 4786 y Fl(j)26 b(\024)f Fp(\032)909 4801 y Fn(k)952 4786 y Fq(,)30 b(1)c Fl(\024)f Fp(k)j Fl(\024)d Fp(p)p Fq(.)2340 b Fc(2)75 5117 y Fr(3)135 b(Summabilit)l(y)46 b(of)f(formal)h(solutions)75 5320 y Fq(All)32 b(the)g(results)f(of)h (this)f(section)i(can)f(b)s(e)f(easily)h(extended)g(to)g(the)g(case)g (of)g(sev)m(eral)h(complex)f(v)-5 b(ariables)32 b Fp(t)p Fq(,)75 5433 y Fp(z)117 5447 y Fo(1)157 5433 y Fp(;)15 b(:)g(:)g(:)h(;)f(z)400 5447 y Fn(n)448 5433 y Fq(,)26 b(but)f(for)g(sak)m(e)h(of)g(simplicit)m(y)-8 b(,)28 b(w)m(e)e(consider)f(no)m(w)g(partial)h(di\013eren)m(tial)h(equations)f (with)f(constan)m(t)75 5546 y(co)s(e\016cien)m(ts)75 5740 y(\(1\))1209 b Fp(P)13 b Fq(\()p Fp(@)1553 5754 y Fn(t)1583 5740 y Fp(;)i(@)1671 5754 y Fn(z)1704 5763 y Fk(1)1743 5740 y Fp(;)g(@)1831 5754 y Fn(z)1864 5763 y Fk(2)1904 5740 y Fq(\))p Fp(u)p Fq(\()p Fp(t;)g(z)2141 5754 y Fo(1)2181 5740 y Fp(;)g(z)2263 5754 y Fo(2)2303 5740 y Fq(\))26 b(=)f(0)p Fp(;)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 75 100 a Fq(6)75 399 y(where)27 b Fp(P)38 b Fl(2)25 b Fe(C)p Fq([)p Fp(@)656 413 y Fn(t)686 399 y Fp(;)15 b(@)774 413 y Fn(z)807 422 y Fk(1)846 399 y Fp(;)g(@)934 413 y Fn(z)967 422 y Fk(2)1006 399 y Fq(])28 b(is)f(a)h(non)e(trivial)j(p)s(olynomial)e(in)g(three)h(v)-5 b(ariables)28 b(with)f(co)s(e\016cien)m(ts)i(in)e Fe(C)p Fq(.)39 b(W)-8 b(e)75 511 y(assume)30 b(moreo)m(v)m(er)i(that)f Fp(P)43 b Fq(can)31 b(b)s(e)f(written)g(in)g(the)h(form)905 760 y Fp(P)13 b Fq(\()p Fp(@)1059 774 y Fn(t)1089 760 y Fp(;)i(@)1177 774 y Fn(z)1210 783 y Fk(1)1249 760 y Fp(;)g(@)1337 774 y Fn(z)1370 783 y Fk(2)1410 760 y Fq(\))25 b(=)g Fp(@)1614 774 y Fn(t)1644 722 y(\024)1689 760 y Fp(p)p Fq(\()p Fp(@)1818 774 y Fn(z)1851 783 y Fk(1)1890 760 y Fp(;)15 b(@)1978 774 y Fn(z)2011 783 y Fk(2)2050 760 y Fq(\))21 b Fl(\000)2242 646 y Fn(\024)2197 674 y Fh(X)2198 868 y Fn(\027)t Fo(=1)2343 760 y Fp(@)2391 774 y Fn(t)2421 722 y(\024)p Fd(\000)p Fn(\027)2560 760 y Fp(p)2606 774 y Fn(\027)2649 760 y Fq(\()p Fp(@)2732 774 y Fn(z)2765 783 y Fk(1)2804 760 y Fp(;)15 b(@)2892 774 y Fn(z)2925 783 y Fk(2)2964 760 y Fq(\))p Fp(;)75 1037 y Fq(where)211 1223 y Fl(\017)46 b Fp(p)p Fq(\()p Fp(w)448 1237 y Fo(1)488 1223 y Fp(;)15 b(w)593 1237 y Fo(2)633 1223 y Fq(\))25 b Fl(2)g Fe(C)p Fq([)p Fp(w)935 1237 y Fo(1)975 1223 y Fp(;)15 b(w)1080 1237 y Fo(2)1120 1223 y Fq(])26 b(is)g(a)g(p)s(olynomial)h(for)e(whic)m(h)h(there)g (exists)h Fp(R)2729 1237 y Fo(0)2795 1223 y Fq(suc)m(h)e(that)i(for)f (all)h Fl(j)p Fp(w)3535 1237 y Fo(1)3574 1223 y Fl(j)f(\025)f Fp(R)3790 1237 y Fo(0)3829 1223 y Fq(,)302 1336 y Fl(j)p Fp(w)392 1350 y Fo(2)432 1336 y Fl(j)h(\025)f Fp(R)648 1350 y Fo(0)687 1336 y Fq(,)31 b(w)m(e)f(ha)m(v)m(e)i Fp(p)p Fq(\()p Fp(w)1232 1350 y Fo(1)1272 1336 y Fp(;)15 b(w)1377 1350 y Fo(2)1416 1336 y Fq(\))26 b Fl(6)p Fq(=)f(0,)211 1523 y Fl(\017)46 b Fp(\024)34 b Fq(is)f(an)g(in)m(teger,)j Fp(p)987 1537 y Fn(\027)1030 1523 y Fq(,)e(1)c Fl(\024)g Fp(\027)35 b Fl(\024)30 b Fp(\024)p Fq(,)k(are)g(p)s(olynomials)f(in)g (t)m(w)m(o)i(v)-5 b(ariables)34 b(and)e Fp(p)3109 1537 y Fn(\024)3187 1523 y Fq(is)h(not)g(iden)m(tically)302 1636 y(equal)e(to)g(zero.)75 1821 y(W)-8 b(e)32 b(consider)e(the)g(c)m (haracteristic)k(equation)d(of)f(\(1\),)75 2093 y(\()p Fp(C)7 b Fq(\))688 b Fp(P)13 b Fq(\()p Fp(\025;)i(w)1169 2107 y Fo(1)1209 2093 y Fp(;)g(w)1314 2107 y Fo(2)1354 2093 y Fq(\))26 b(=)f Fp(p)p Fq(\()p Fp(w)1657 2107 y Fo(1)1696 2093 y Fp(;)15 b(w)1801 2107 y Fo(2)1841 2093 y Fq(\))p Fp(\025)1929 2055 y Fn(\024)1995 2093 y Fl(\000)2131 1979 y Fn(\024)2085 2006 y Fh(X)2087 2200 y Fn(\027)t Fo(=1)2232 2093 y Fp(p)2278 2107 y Fn(\027)2321 2093 y Fq(\()p Fp(w)2421 2107 y Fo(1)2461 2093 y Fp(;)g(w)2566 2107 y Fo(2)2606 2093 y Fq(\))p Fp(\025)2694 2055 y Fn(\024)p Fd(\000)p Fn(\027)2858 2093 y Fq(=)25 b(0)p Fp(;)75 2375 y Fq(with)30 b Fl(j)p Fp(w)372 2389 y Fo(1)412 2375 y Fl(j)25 b(\025)g Fp(R)627 2389 y Fo(0)697 2375 y Fq(and)30 b Fl(j)p Fp(w)964 2389 y Fo(2)1004 2375 y Fl(j)25 b(\025)g Fp(R)1219 2389 y Fo(0)1259 2375 y Fq(.)40 b(In)30 b(the)g(follo)m (wing,)i(w)m(e)f(put)49 b(~)-64 b Fp(w)2369 2389 y Fn(j)2431 2375 y Fq(=)25 b(1)p Fp(=w)2682 2389 y Fn(j)2719 2375 y Fq(,)31 b Fp(j)g Fq(=)25 b(1)p Fp(;)15 b Fq(2.)216 2537 y(F)-8 b(or)29 b Fp(m)457 2551 y Fo(1)496 2537 y Fq(,)g Fp(m)630 2551 y Fo(2)697 2537 y Fq(su\016cien)m(tly)f(large)h (in)m(tegers,)h(w)m(e)f(m)m(ultiply)f(the)g(equation)h(\()p Fp(C)7 b Fq(\))28 b(b)m(y)f(the)i(rational)g(function)1167 2739 y Fp(N)10 b Fq(\()19 b(~)-64 b Fp(w)1350 2753 y Fo(1)1390 2739 y Fp(;)34 b Fq(~)-64 b Fp(w)1495 2753 y Fo(2)1535 2739 y Fq(\))25 b(=)g(\()19 b(~)-64 b Fp(w)1793 2698 y Fn(m)1855 2707 y Fk(1)1791 2765 y Fo(1)1914 2739 y Fq(~)g Fp(w)1962 2698 y Fn(m)2024 2707 y Fk(2)1960 2765 y Fo(2)2064 2739 y Fq(\))2099 2701 y Fn(\024)2144 2739 y Fp(=p)p Fq(\(1)p Fp(=)20 b Fq(~)-65 b Fp(w)2425 2753 y Fo(1)2466 2739 y Fp(;)15 b Fq(1)p Fp(=)20 b Fq(~)-65 b Fp(w)2661 2753 y Fo(2)2702 2739 y Fq(\))p Fp(;)75 2941 y Fq(and)30 b(w)m(e)h(get)g(the)g(follo)m(wing)h(equation)821 3207 y(\()19 b(~)-64 b Fp(w)923 3166 y Fn(m)985 3175 y Fk(1)921 3233 y Fo(1)1044 3207 y Fq(~)g Fp(w)1092 3166 y Fn(m)1154 3175 y Fk(2)1090 3233 y Fo(2)1194 3207 y Fq(\))1229 3169 y Fn(\024)1274 3207 y Fp(\025)1327 3169 y Fn(\024)1392 3207 y Fl(\000)1528 3093 y Fn(\024)1483 3120 y Fh(X)1484 3315 y Fn(\027)t Fo(=1)1614 3207 y Fq(\()19 b(~)-64 b Fp(w)1716 3166 y Fn(m)1778 3175 y Fk(1)1714 3233 y Fo(1)1837 3207 y Fq(~)g Fp(w)1885 3166 y Fn(m)1947 3175 y Fk(2)1883 3233 y Fo(2)1987 3207 y Fq(\))2022 3169 y Fn(\024)p Fd(\000)p Fo(1)2244 3145 y Fq(~)-52 b Fp(p)2283 3159 y Fn(\027)2325 3145 y Fq(\()19 b(~)-64 b Fp(w)2425 3159 y Fo(1)2465 3145 y Fp(;)34 b Fq(~)-64 b Fp(w)2570 3159 y Fo(2)2610 3145 y Fq(\))p 2167 3186 548 4 v 2167 3269 a Fp(p)p Fq(\(1)p Fp(=)20 b Fq(~)-65 b Fp(w)2403 3283 y Fo(1)2444 3269 y Fp(;)15 b Fq(1)p Fp(=)20 b Fq(~)-65 b Fp(w)2639 3283 y Fo(2)2680 3269 y Fq(\))2725 3207 y Fp(\025)2778 3169 y Fn(\024)p Fd(\000)p Fn(\027)2942 3207 y Fq(=)25 b(0)p Fp(;)75 3484 y Fq(where)38 b(~)-53 b Fp(p)384 3498 y Fn(\027)427 3484 y Fq(,)30 b(1)c Fl(\024)f Fp(\027)31 b Fl(\024)25 b Fp(\024)p Fq(,)31 b(are)g(p)s(olynomials)f (in)g(the)h(t)m(w)m(o)h(v)-5 b(ariables)49 b(~)-63 b Fp(w)2463 3498 y Fo(1)2532 3484 y Fq(and)49 b(~)-64 b Fp(w)2774 3498 y Fo(2)2814 3484 y Fq(.)216 3646 y(Making)31 b(the)g(c)m(hange)g(of)g(v)-5 b(ariable)31 b Fp(\025)25 b Fq(=)1624 3622 y(~)1620 3646 y Fp(\025=)20 b Fq(~)-65 b Fp(w)1785 3606 y Fn(m)1847 3615 y Fk(1)1783 3672 y Fo(1)1906 3646 y Fq(~)i Fp(w)1955 3606 y Fn(m)2017 3615 y Fk(2)1953 3672 y Fo(2)2056 3646 y Fq(,)31 b(w)m(e)g(get)g(the)g (equation)75 3921 y(\()131 3898 y(~)110 3921 y Fp(C)7 b Fq(\))766 3898 y(~)745 3921 y Fp(P)13 b Fq(\()855 3897 y(~)851 3921 y Fp(\025)q(;)34 b Fq(~)-64 b Fp(w)1010 3935 y Fo(1)1050 3921 y Fp(;)34 b Fq(~)-64 b Fp(w)1155 3935 y Fo(2)1195 3921 y Fq(\))25 b(=)1355 3897 y(~)1351 3921 y Fp(\025)1404 3883 y Fn(\024)1469 3921 y Fl(\000)1605 3807 y Fn(\024)1560 3834 y Fh(X)1561 4029 y Fn(\027)t Fo(=1)1691 3921 y Fq(\()19 b(~)-64 b Fp(w)1793 3880 y Fn(m)1855 3889 y Fk(1)1791 3947 y Fo(1)1914 3921 y Fq(~)g Fp(w)1962 3880 y Fn(m)2024 3889 y Fk(2)1960 3947 y Fo(2)2064 3921 y Fq(\))2099 3883 y Fn(\027)t Fd(\000)p Fo(1)2320 3859 y Fq(~)-53 b Fp(p)2358 3873 y Fn(\027)2401 3859 y Fq(\()19 b(~)-64 b Fp(w)2501 3873 y Fo(1)2541 3859 y Fp(;)34 b Fq(~)-64 b Fp(w)2646 3873 y Fo(2)2686 3859 y Fq(\))p 2243 3900 V 2243 3983 a Fp(p)p Fq(\(1)p Fp(=)19 b Fq(~)-64 b Fp(w)2479 3997 y Fo(1)2519 3983 y Fp(;)15 b Fq(1)p Fp(=)20 b Fq(~)-65 b Fp(w)2714 3997 y Fo(2)2755 3983 y Fq(\))2804 3897 y(~)2800 3921 y Fp(\025)2853 3883 y Fn(\024)p Fd(\000)p Fn(\027)3018 3921 y Fq(=)24 b(0)p Fp(:)75 4219 y Fq(for)29 b(\()19 b(~)-64 b Fp(w)313 4233 y Fo(1)353 4219 y Fp(;)34 b Fq(~)-64 b Fp(w)458 4233 y Fo(2)498 4219 y Fq(\))25 b Fl(2)g Fp(D)719 4237 y Fo(1)p Fn(=R)842 4246 y Fk(0)899 4219 y Fl(\002)17 b Fp(D)1062 4237 y Fo(1)p Fn(=R)1185 4246 y Fk(0)1225 4219 y Fq(.)40 b(In)28 b(the)i(follo)m(wing,)h(w)m(e)e(use)g(the)g(notation)2802 4196 y(_)2771 4219 y Fp(D)2846 4233 y Fn(\032)2912 4219 y Fq(=)c Fl(f)p Fp(z)30 b Fl(2)25 b Fe(C)g Fq(:)g(0)h Fp(<)f Fl(j)p Fp(z)t Fl(j)h Fp(<)f(\032)p Fl(g)p Fq(.)216 4396 y(W)-8 b(e)31 b(denote)f(b)m(y)g Fp(D)881 4409 y Fo(~)865 4426 y Fn(P)954 4396 y Fq(the)g(discriminan)m(t)f(of)h(the)g (p)s(olynomial)2381 4373 y(~)2360 4396 y Fp(P)43 b Fq(in)29 b(the)h(v)-5 b(ariable)3067 4372 y(~)3063 4396 y Fp(\025)p Fq(.)40 b(In)29 b(this)h(pap)s(er,)f(w)m(e)75 4509 y(assume)h(that)75 4710 y(\()p Fp(C)7 b(d)p Fq(1\))1333 b Fp(D)1733 4723 y Fo(~)1717 4740 y Fn(P)1776 4710 y Fq(\()19 b(~)-64 b Fp(w)1876 4724 y Fo(1)1916 4710 y Fp(;)34 b Fq(~)-64 b Fp(w)2021 4724 y Fo(2)2061 4710 y Fq(\))25 b Fl(6)p Fq(=)g(0)p Fp(;)75 4921 y Fq(for)30 b(all)h(\()19 b(~)-64 b Fp(w)440 4935 y Fo(1)480 4921 y Fp(;)34 b Fq(~)-64 b Fp(w)585 4935 y Fo(2)625 4921 y Fq(\))26 b Fl(2)803 4898 y Fq(_)772 4921 y Fp(D)847 4940 y Fo(1)p Fn(=R)970 4949 y Fk(0)1030 4921 y Fl(\002)1152 4898 y Fq(_)1121 4921 y Fp(D)1196 4940 y Fo(1)p Fn(=R)1319 4949 y Fk(0)1358 4921 y Fq(.)216 5083 y(Due)g(to)g(the)f(result)g(of)g(J.)h(Cel)f(\(see) h([4]\),)i(there)d(exist)h(t)m(w)m(o)h(p)s(ositiv)m(e)f(in)m(tegers)h Fp(q)2927 5097 y Fo(1)2966 5083 y Fq(,)f Fp(q)3058 5097 y Fo(2)3122 5083 y Fq(and)f(a)g(holomorphic)75 5196 y(function)30 b Fp(H)37 b Fq(on)31 b Fp(D)746 5224 y Fo(\(1)p Fn(=R)896 5233 y Fk(0)932 5224 y Fo(\))959 5202 y Fk(1)p Fg(=q)1049 5217 y Fk(1)1113 5196 y Fl(\002)20 b Fp(D)1279 5224 y Fo(\(1)p Fn(=R)1429 5233 y Fk(0)1465 5224 y Fo(\))1492 5202 y Fk(1)p Fg(=q)1582 5217 y Fk(2)1656 5196 y Fq(suc)m(h)30 b(that)1260 5403 y(~)1240 5426 y Fp(P)13 b Fq(\()p Fp(H)7 b Fq(\(\()20 b(~)-65 b Fp(w)1564 5440 y Fo(1)1604 5426 y Fq(\))1639 5388 y Fo(1)p Fn(=q)1741 5397 y Fk(1)1781 5426 y Fp(;)15 b Fq(\()k(~)-64 b Fp(w)1921 5440 y Fo(2)1961 5426 y Fq(\))1996 5388 y Fo(1)p Fn(=q)2098 5397 y Fk(2)2137 5426 y Fq(\))p Fp(;)34 b Fq(~)-64 b Fp(w)2277 5440 y Fo(1)2317 5426 y Fp(;)34 b Fq(~)-64 b Fp(w)2422 5440 y Fo(2)2462 5426 y Fq(\))26 b(=)f(0)p Fp(:)75 5627 y Ff(Remark.)62 b Fq(In)36 b(fact,)41 b(the)c(result)g(of)h(J.)f(Cel)h (sho)m(ws)f(that)g(there)h(are)g Fp(\024)f Fq(suc)m(h)g(holomorphic)h (functions)f Fp(H)75 5740 y Fq(satisfying)31 b(the)g(latter)g (equation.)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3809 100 a Fq(7)75 399 y Ff(Prop)s(osition)36 b(3)46 b Fm(Under)33 b(the)g(c)-5 b(onditions)34 b Fq(\()p Fp(C)7 b(d)p Fq(1\))p Fm(,)34 b(ther)-5 b(e)33 b(exists)g(a)g(function) g Fp(\025)f Fm(of)h(the)g(form)1189 574 y Fp(\025)p Fq(\()p Fp(w)1342 588 y Fo(1)1382 574 y Fp(;)15 b(w)1487 588 y Fo(2)1526 574 y Fq(\))26 b(=)f Fp(w)1750 533 y Fn(m)1812 542 y Fk(1)1748 600 y Fo(1)1852 574 y Fp(w)1919 533 y Fn(m)1981 542 y Fk(2)1917 600 y Fo(2)2020 574 y Fp(H)7 b Fq(\()p Fp(w)2205 526 y Fd(\000)p Fo(1)p Fn(=q)2362 535 y Fk(1)2203 600 y Fo(1)2402 574 y Fp(;)15 b(w)2509 526 y Fd(\000)p Fo(1)p Fn(=q)2666 535 y Fk(2)2507 600 y Fo(2)2706 574 y Fq(\))75 733 y Fm(wher)-5 b(e)34 b Fp(H)39 b Fm(is)33 b(a)g(holomorphic)j(function)c(on)i Fp(D)1704 762 y Fo(\(1)p Fn(=R)1854 771 y Fk(0)1890 762 y Fo(\))1917 739 y Fk(1)p Fg(=q)2007 754 y Fk(1)2071 733 y Fl(\002)20 b Fp(D)2237 762 y Fo(\(1)p Fn(=R)2387 771 y Fk(0)2423 762 y Fo(\))2450 739 y Fk(1)p Fg(=q)2540 754 y Fk(2)2616 733 y Fm(such)33 b(that)1479 901 y Fp(P)13 b Fq(\()p Fp(\025)p Fq(\()p Fp(w)1738 915 y Fo(1)1778 901 y Fp(;)i(w)1883 915 y Fo(2)1923 901 y Fq(\))p Fp(;)g(w)2063 915 y Fo(1)2104 901 y Fp(;)g(w)2209 915 y Fo(2)2248 901 y Fq(\))26 b(=)f(0)75 1061 y Fm(for)31 b Fl(j)p Fp(w)308 1075 y Fo(1)347 1061 y Fl(j)26 b(\025)f Fp(R)563 1075 y Fo(0)602 1061 y Fm(,)31 b Fl(j)p Fp(w)751 1075 y Fo(2)790 1061 y Fl(j)26 b(\025)f Fp(R)1006 1075 y Fo(0)1045 1061 y Fm(.)41 b(In)30 b(p)-5 b(articular,)33 b(the)d(function)g Fp(\025)g Fm(satis\014es)h(the)f(assumption)i Ff(A)e Fm(of)g(the)h(se)-5 b(ction)75 1174 y(1.)49 b(So)36 b(that)g(the)f (formal)i(series)e Fq(\(2\))h Fm(c)-5 b(onstructe)g(d)37 b(in)e(the)g(pr)-5 b(op)g(osition)39 b(2)c(with)h(the)f(help)h(of)g Fp(\025)e Fm(is)h(a)h(formal)75 1287 y(solution)k(of)f Fq(\(1\))g Fm(and)h(satis\014es)g(the)f(Gevr)-5 b(ey)39 b(estimates)g(of)g(the)g(the)-5 b(or)g(em)41 b(1.)60 b(In)39 b(fact,)h(due)f(to)g(the)g(upp)-5 b(er)75 1399 y(mentione)g(d)30 b(r)-5 b(emark,)31 b(one)e(c)-5 b(an)29 b(c)-5 b(onstruct)30 b Fp(\024)f Fm(such)g(functions)g Fp(\025)g Fm(and)h(ther)-5 b(efor)g(e)30 b Fp(\024)f Fm(formal)i(series)e Fq(\(2\))g Fm(which)75 1512 y(ar)-5 b(e)33 b(formal)i(solutions)f(of)f Fq(\(1\))g Fm(and)h(satisfy)f(the)g (Gevr)-5 b(ey)33 b(estimates)h(of)f(the)g(the)-5 b(or)g(em)35 b(1.)216 1663 y Fq(In)41 b(order)g(to)i(study)e(the)h(summabilit)m(y)g (prop)s(erties)f(of)h(the)g(formal)g(p)s(o)m(w)m(er)g(series)g (obtained)g(in)f(the)75 1776 y(previous)32 b(sections,)h(w)m(e)g(will)f (mak)m(e)h(an)f(assumption)g(on)g(the)g(holomorphic)g(function)g Fp(H)39 b Fq(obtained)32 b(in)g(the)75 1889 y(prop)s(osition)e(3.)41 b(F)-8 b(rom)31 b(no)m(w)f(on,)h(w)m(e)g(assume)f(that)h(the)f (function)h Fp(H)37 b Fq(can)30 b(b)s(e)g(factorized)i(in)e(the)h(form) 75 2048 y(\()p Fp(C)7 b(d)p Fq(2\))807 b Fp(H)7 b Fq(\()p Fp(Z)1296 2062 y Fo(1)1336 2048 y Fp(;)15 b(Z)1438 2062 y Fo(2)1478 2048 y Fq(\))25 b(=)g Fp(Z)1703 2008 y Fn(\013)1748 2017 y Fk(1)1696 2074 y Fo(1)1787 2048 y Fp(Z)1856 2008 y Fn(\013)1901 2017 y Fk(2)1849 2074 y Fo(2)1939 2048 y Fq(\()p Fp(\025)2027 2062 y Fo(0)2087 2048 y Fq(+)20 b Fp(Z)2240 2062 y Fo(1)2280 2048 y Fp(Z)2342 2062 y Fo(2)2405 2025 y Fq(~)2381 2048 y Fp(H)7 b Fq(\()p Fp(Z)2561 2062 y Fo(1)2601 2048 y Fp(;)15 b(Z)2703 2062 y Fo(2)2743 2048 y Fq(\)\))75 2208 y(where)369 2185 y(~)346 2208 y Fp(H)45 b Fq(is)38 b(a)g(holomorphic)h(function)e(on)h Fp(D)1750 2236 y Fo(\(1)p Fn(=R)1900 2245 y Fk(0)1937 2236 y Fo(\))1964 2214 y Fk(1)p Fg(=q)2054 2229 y Fk(1)2123 2208 y Fl(\002)25 b Fp(D)2294 2236 y Fo(\(1)p Fn(=R)2444 2245 y Fk(0)2480 2236 y Fo(\))2507 2214 y Fk(1)p Fg(=q)2597 2229 y Fk(2)2641 2208 y Fq(,)40 b Fp(\025)2759 2222 y Fo(0)2837 2208 y Fl(6)p Fq(=)e(0)g(and)g 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Fk(1)1346 2915 y Fn(=q)1413 2924 y Fk(1)1288 2989 y Fo(1)1451 2963 y Fp(w)1518 2915 y Fn(l)1539 2924 y Fk(2)1574 2915 y Fn(=q)1641 2924 y Fk(2)1516 2989 y Fo(2)1680 2963 y Fq(\()p Fp(\025)1768 2977 y Fo(0)1828 2963 y Fq(+)20 b Fp(w)1986 2915 y Fd(\000)p Fo(1)p Fn(=q)2143 2924 y Fk(1)1984 2989 y Fo(1)2182 2963 y Fp(w)2249 2915 y Fd(\000)p Fo(1)p Fn(=q)2406 2924 y Fk(1)2247 2989 y Fo(2)2469 2940 y Fq(~)2445 2963 y Fp(H)7 b Fq(\()p Fp(w)2630 2915 y Fd(\000)p Fo(1)p Fn(=q)2787 2924 y Fk(1)2628 2989 y Fo(1)2827 2963 y Fp(;)15 b(w)2934 2915 y Fd(\000)p Fo(1)p Fn(=q)3091 2924 y Fk(2)2932 2989 y Fo(2)3131 2963 y Fq(\)\))75 3123 y(where)362 3100 y(~)338 3123 y Fp(H)37 b Fq(is)31 b(a)f(holomorphic)h(function)f(on)g Fp(D)1696 3151 y Fo(\(1)p Fn(=R)1846 3160 y Fk(0)1882 3151 y Fo(\))1909 3129 y Fk(1)p Fg(=q)1999 3144 y Fk(1)2063 3123 y Fl(\002)20 b Fp(D)2229 3151 y Fo(\(1)p Fn(=R)2379 3160 y Fk(0)2415 3151 y Fo(\))2442 3129 y Fk(1)p Fg(=q)2532 3144 y Fk(2)2607 3123 y Fq(and)1503 3347 y(1)26 b 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y Fd(\000)p Fn(\026)2587 3888 y Fk(1)2623 3879 y Fn(=q)2690 3888 y Fk(1)2488 3953 y Fo(1)2728 3927 y Fp(w)2795 3879 y Fd(\000)p Fn(\026)2892 3888 y Fk(2)2928 3879 y Fn(=q)2995 3888 y Fk(2)2793 3953 y Fo(2)75 4090 y Fq(for)30 b Fl(j)p Fp(w)304 4104 y Fo(1)344 4090 y Fl(j)25 b(\025)g Fp(R)559 4104 y Fo(0)629 4090 y Fq(and)30 b Fl(j)p Fp(w)896 4104 y Fo(2)936 4090 y Fl(j)25 b(\025)g Fp(R)1151 4104 y Fo(0)1191 4090 y Fq(.)75 4238 y Ff(Example)35 b(1)30 b Fq(Let)h(the)g(c)m(haracteristic)i (equation)e(of)g(\(1\))g(b)s(e)f(of)g(the)h(form)75 4467 y(\()p Fp(E)5 b Fq(1\))767 b Fp(P)13 b Fq(\()p Fp(\025;)i(w)1293 4481 y Fo(1)1333 4467 y Fp(;)g(w)1438 4481 y Fo(2)1478 4467 y Fq(\))25 b(=)g Fp(p)p Fq(\()p Fp(w)1780 4481 y Fo(1)1820 4467 y Fp(w)1885 4481 y Fo(2)1924 4467 y Fq(\))p Fp(\025)2012 4429 y Fn(\024)2078 4467 y Fl(\000)2214 4353 y Fn(\024)2169 4380 y Fh(X)2170 4575 y Fn(\027)t Fo(=1)2315 4467 y Fp(p)2361 4481 y Fn(\027)2404 4467 y Fq(\()p Fp(w)2504 4481 y Fo(1)2544 4467 y Fp(w)2609 4481 y Fo(2)2649 4467 y Fq(\))p Fp(\025)2737 4429 y Fn(\024)p Fd(\000)p Fn(\027)2876 4467 y Fp(;)75 4707 y Fq(where)30 b Fp(p)p Fq(,)h Fp(p)486 4721 y Fn(\027)555 4707 y Fl(2)25 b Fe(C)p Fq([)p Fp(X)7 b Fq(],)31 b(1)c Fl(\024)e Fp(\027)31 b Fl(\024)26 b Fp(\024)31 b Fq(and)f Fp(p)1542 4721 y Fn(\024)1617 4707 y Fq(is)h(a)g(not)g(iden)m(tically)i(equal)e(to)g (zero.)43 b(F)-8 b(rom)31 b(a)g(classical)i(result)75 4820 y(\(see)e([1]\),)h(there)f(exists)g(an)f(algebraic)i(function)e Fp(\034)10 b Fq(\()p Fp(z)t Fq(\))32 b(with)e(expansion)g(of)g(the)h (form)1314 4979 y Fp(\034)10 b Fq(\()p Fp(z)t Fq(\))26 b(=)f Fp(z)1648 4942 y Fn(l)q(=q)1743 4979 y Fq(\()p Fp(\025)1831 4993 y Fo(0)1891 4979 y Fq(+)20 b Fp(z)2028 4942 y Fd(\000)p Fo(1)p Fn(=q)2216 4956 y Fq(~)2192 4979 y Fp(H)7 b Fq(\()p Fp(z)2356 4942 y Fd(\000)p Fo(1)p Fn(=q)2520 4979 y Fq(\)\))p Fp(;)75 5139 y Fq(where)362 5116 y(~)338 5139 y Fp(H)37 b Fq(is)31 b(holomorphic)f(on)g(a)h(disc)f Fp(D)1522 5164 y Fo(\(1)p Fn(=R)1672 5173 y Fk(0)1708 5164 y Fo(\))1735 5145 y Fk(1)p Fg(=q)1837 5139 y Fq(,)g Fp(R)1961 5153 y Fo(0)2026 5139 y Fp(>)25 b Fq(0,)31 b(whic)m(h)f(satis\014es)h(the)f(equation)1414 5387 y Fp(p)p Fq(\()p Fp(z)t Fq(\))p Fp(\034)1626 5350 y Fn(\024)1692 5387 y Fl(\000)1828 5274 y Fn(\024)1783 5301 y Fh(X)1784 5495 y Fn(\027)t Fo(=1)1929 5387 y Fp(p)1975 5401 y Fn(\027)2018 5387 y Fq(\()p Fp(z)t Fq(\))p Fp(\034)2184 5350 y Fn(\024)p Fd(\000)p Fn(\027)2349 5387 y Fq(=)25 b(0)p Fp(:)75 5627 y Fq(W)-8 b(e)37 b(require)e(moreo)m(v)m(er)j(that)e Fp(l)r(=q)i(>)c Fq(1)p Fp(=)p Fq(2.)58 b(If)35 b(w)m(e)i(put)e Fp(z)j Fq(=)c Fp(w)2282 5641 y Fo(1)2322 5627 y Fp(w)2387 5641 y Fo(2)2426 5627 y Fq(,)k(the)e(function)f Fp(\025)p Fq(\()p Fp(w)3165 5641 y Fo(1)3205 5627 y Fp(;)15 b(w)3310 5641 y Fo(2)3350 5627 y Fq(\))35 b(=)f Fp(\034)10 b Fq(\()p Fp(w)3675 5641 y Fo(1)3715 5627 y Fp(w)3780 5641 y Fo(2)3819 5627 y Fq(\))75 5740 y(satis\014es)31 b(\()p Fp(C)7 b(d)p Fq(21\))32 b(and)d(the)i(equation)g Fp(P)13 b Fq(\()p Fp(\025)p Fq(\()p Fp(w)1684 5754 y Fo(1)1724 5740 y Fp(;)i(w)1829 5754 y Fo(2)1869 5740 y Fq(\))p Fp(;)g(w)2009 5754 y Fo(1)2050 5740 y Fp(;)g(w)2155 5754 y Fo(2)2194 5740 y Fq(\))26 b(=)f(0)31 b(for)f Fl(j)p Fp(w)2656 5754 y Fo(1)2696 5740 y Fl(j)25 b(\025)g Fp(R)2911 5754 y Fo(0)2981 5740 y Fq(and)30 b Fl(j)p Fp(w)3248 5754 y Fo(2)3287 5740 y Fl(j)c(\025)f Fp(R)3503 5754 y Fo(0)3542 5740 y Fq(.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 75 100 a Fq(8)75 399 y Fb(3.1)112 b(In)m(tegral)38 b(represen)m(tations)75 570 y Fq(In)30 b(the)g(follo)m(wing,)i(w)m(e)f(put)f(for)g Fp(j)h Fl(\025)25 b Fq(0,)888 845 y Fp(g)931 859 y Fn(j)968 845 y Fq(\()p Fp(z)1045 859 y Fo(1)1085 845 y Fp(;)15 b(z)1167 859 y Fo(2)1207 845 y Fq(\))26 b(=)1460 731 y Fd(1)1430 759 y Fh(X)1364 953 y Fn(\026)1406 962 y Fk(1)1441 953 y Fn(;\026)1503 962 y Fk(2)1537 953 y Fo(=0)1643 845 y Fp(c)1682 859 y Fn(j)t(\026)1757 868 y Fk(1)1791 859 y Fn(\026)1833 868 y Fk(2)2228 783 y Fp(z)2274 736 y Fn(\026)2316 745 y Fk(1)2352 736 y Fn(=q)2419 745 y Fk(1)2270 809 y Fo(1)2457 783 y Fp(z)2503 736 y Fn(\026)2545 745 y Fk(2)2580 736 y Fn(=q)2647 745 y Fk(2)2499 809 y Fo(2)p 1883 824 1149 4 v 1883 907 a Fq(\000\(1)20 b(+)g(\()p Fp(\026)2221 921 y Fo(1)2261 907 y Fp(=q)2347 921 y Fo(1)2386 907 y Fq(\)\)\000\(1)i(+)e(\()p Fp(\026)2796 921 y Fo(2)2835 907 y Fp(=q)2921 921 y Fo(2)2961 907 y Fq(\)\))75 1141 y(for)32 b(all)h(\()p Fp(z)421 1155 y Fo(1)461 1141 y Fp(;)15 b(z)543 1155 y Fo(2)583 1141 y Fq(\))28 b Fl(2)g Fp(D)810 1155 y Fn(\032)846 1164 y Fk(1)906 1141 y Fl(\002)21 b Fp(D)1073 1155 y Fn(\032)1109 1164 y Fk(2)1148 1141 y Fq(.)46 b(The)31 b(follo)m(wing)j(prop)s(osition)d(giv)m(es)j(an)d (in)m(tegral)j(represen)m(tation)f(of)f Fp(u)3710 1155 y Fn(j)3779 1141 y Fq(in)75 1254 y(terms)e(of)h(\010)f(and)g Fp(g)748 1268 y Fn(j)784 1254 y Fq(.)75 1440 y Ff(Prop)s(osition)36 b(4)46 b Fm(F)-7 b(or)34 b(al)5 b(l)33 b Fp(j)e Fl(\025)25 b Fq(0)p Fm(,)32 b Fq(\()p Fp(z)1384 1454 y Fo(1)1425 1440 y Fp(;)15 b(z)1507 1454 y Fo(2)1547 1440 y Fq(\))25 b Fl(2)g Fp(D)1768 1454 y Fn(\032)1804 1463 y Fk(1)1863 1440 y Fl(\002)20 b Fp(D)2029 1454 y Fn(\032)2065 1463 y Fk(2)2105 1440 y Fm(,)107 1714 y Fp(u)159 1728 y Fn(j)195 1714 y Fq(\()p Fp(z)272 1728 y Fo(1)312 1714 y Fp(;)15 b(z)394 1728 y Fo(2)434 1714 y Fq(\))26 b(=)601 1640 y(\000\(1)21 b(+)f Fp(l)879 1607 y Fo(1)877 1666 y Fn(j)918 1640 y Fq(\)\000\(1)h(+)f Fp(l)1231 1607 y Fo(2)1229 1666 y Fn(j)1270 1640 y Fq(\))p 601 1693 705 4 v 752 1776 a(\(2)p Fp(\031)s(i)p Fq(\))953 1750 y Fo(2)994 1776 y Fp(q)1035 1790 y Fo(1)1074 1776 y Fp(q)1115 1790 y Fo(2)1316 1714 y Fp(@)1364 1728 y Fn(z)1397 1737 y Fk(1)1436 1714 y Fp(@)1484 1728 y Fn(z)1517 1737 y Fk(2)1571 1590 y Fh(I)1621 1797 y Fn(q)1653 1806 y Fk(1)1687 1797 y Fn(C)1737 1805 y Fg(\032)1769 1820 y Fk(1)1827 1590 y Fh(I)1878 1797 y Fn(q)1910 1806 y Fk(2)1944 1797 y Fn(C)1994 1805 y Fg(\032)2026 1820 y Fk(2)2084 1714 y Fq(\010)g Fl(\003)h Fp(g)2279 1728 y Fn(j)2315 1714 y Fq(\()p Fp(w)2415 1728 y Fo(1)2455 1714 y Fp(;)15 b(w)2560 1728 y Fo(2)2600 1714 y Fq(\))2857 1653 y Fp(dw)2969 1667 y Fo(1)p 2645 1693 574 4 v 2645 1801 a Fq(\()p Fp(w)2745 1815 y Fo(1)2806 1801 y Fl(\000)20 b Fp(z)2939 1815 y Fo(1)2978 1801 y Fq(\))3013 1759 y Fo(\(1+)p Fn(l)3152 1736 y Fk(1)3151 1781 y Fg(j)3188 1759 y Fo(\))3450 1653 y Fp(dw)3562 1667 y Fo(2)p 3239 1693 V 3239 1801 a Fq(\()p Fp(w)3339 1815 y Fo(2)3399 1801 y Fl(\000)g Fp(z)3532 1815 y Fo(2)3572 1801 y Fq(\))3607 1759 y Fo(\(1+)p Fn(l)3746 1736 y Fk(2)3745 1781 y Fg(j)3781 1759 y Fo(\))75 1997 y Fm(wher)-5 b(e)34 b Fq(\010)20 b Fl(\003)g Fp(g)526 2011 y Fn(j)563 1997 y Fq(\()p Fp(w)663 2011 y Fo(1)703 1997 y Fp(;)15 b(w)808 2011 y Fo(2)848 1997 y Fq(\))26 b(=)1004 1924 y Fh(R)1065 1950 y Fn(w)1116 1959 y Fk(2)1047 2029 y Fo(0)1170 1924 y Fh(R)1230 1950 y Fn(w)1281 1959 y Fk(1)1213 2029 y Fo(0)1335 1997 y Fq(\010\()p Fp(w)1501 2011 y Fo(1)1560 1997 y Fl(\000)20 b Fp(u)1703 2011 y Fo(1)1743 1997 y Fp(;)15 b(w)1848 2011 y Fo(2)1908 1997 y Fl(\000)20 b Fp(u)2051 2011 y Fo(2)2090 1997 y Fq(\))p Fp(g)2168 2011 y Fn(j)2206 1997 y Fq(\()p Fp(u)2293 2011 y Fo(1)2332 1997 y Fp(;)15 b(u)2424 2011 y Fo(2)2464 1997 y Fq(\))p Fp(du)2598 2011 y Fo(1)2638 1997 y Fp(du)2737 2011 y Fo(2)2777 1997 y Fm(.)75 2232 y Ff(Pro)s(of)31 b Fq(Using)g(the)g(expansion)f(\()p Fp(exp\025)1422 2199 y Fn(j)1459 2232 y Fq(\))h(and)e(the)i(residues)f(theorem,)h(w)m(e)g (ma)m(y)g(expand)f Fp(u)3281 2246 y Fn(j)3317 2232 y Fq(\()p Fp(z)3394 2246 y Fo(1)3434 2232 y Fp(;)15 b(z)3516 2246 y Fo(2)3556 2232 y Fq(\))31 b(in)f(the)75 2345 y(form)75 2623 y(\()p Fp(u)162 2525 y Fa(P)162 2650 y Fn(j)241 2623 y Fq(\))278 b Fp(u)606 2637 y Fn(j)642 2623 y Fq(\()p Fp(z)719 2637 y Fo(1)759 2623 y Fp(;)15 b(z)841 2637 y Fo(2)882 2623 y Fq(\))25 b(=)1038 2536 y Fh(X)1059 2733 y Fd(A)1115 2743 y Fg(j)1185 2623 y Fp(c)1224 2637 y Fn(j)t(\026)1299 2646 y Fk(1)1333 2637 y Fn(\026)1375 2646 y Fk(2)1414 2623 y Fq(\010)1480 2637 y Fn(n)1523 2646 y Fk(1)1558 2637 y Fn(n)1601 2646 y Fk(2)1962 2561 y Fp(n)2017 2575 y Fo(1)2056 2561 y Fq(!)p Fp(n)2136 2575 y Fo(2)2175 2561 y Fq(!)p Fp(z)2246 2499 y Fo(\()p Fn(\026)2315 2508 y Fk(1)2351 2499 y Fn(=q)2418 2508 y Fk(1)2452 2499 y Fo(\)+)p Fn(n)2577 2508 y Fk(1)2612 2499 y Fd(\000)p Fn(l)2689 2475 y Fk(1)2688 2520 y Fg(j)2242 2587 y Fo(1)2727 2561 y Fp(z)2773 2499 y Fo(\()p Fn(\026)2842 2508 y Fk(2)2878 2499 y Fn(=q)2945 2508 y Fk(2)2980 2499 y Fo(\)+)p Fn(n)3105 2508 y Fk(2)3139 2499 y Fd(\000)p Fn(l)3216 2475 y Fk(2)3215 2520 y Fg(j)2769 2587 y Fo(2)p 1649 2602 1919 4 v 1649 2687 a Fq(\000\(1)c(+)f(\()p Fp(\026)1988 2701 y Fo(1)2027 2687 y Fp(=q)2113 2701 y Fo(1)2153 2687 y Fq(\))g(+)g Fp(n)2354 2701 y Fo(1)2414 2687 y Fl(\000)f Fp(l)2533 2655 y Fo(1)2531 2714 y Fn(j)2573 2687 y Fq(\)\000\(1)i(+)f(\()p Fp(\026)2947 2701 y Fo(2)2986 2687 y Fp(=q)3072 2701 y Fo(2)3112 2687 y Fq(\))g(+)g Fp(n)3313 2701 y Fo(2)3373 2687 y Fl(\000)f Fp(l)3492 2655 y Fo(2)3490 2714 y Fn(j)3532 2687 y Fq(\))75 2936 y(for)25 b(all)h(\()p Fp(z)407 2950 y Fo(1)447 2936 y Fp(;)15 b(z)529 2950 y Fo(2)569 2936 y Fq(\))25 b Fl(2)g Fp(D)790 2950 y Fn(\032)826 2959 y Fk(1)875 2936 y Fl(\002)9 b Fp(D)1030 2950 y Fn(\032)1066 2959 y Fk(2)1105 2936 y Fq(,)26 b(where)e Fl(A)1486 2950 y Fn(j)1548 2936 y Fq(=)h Fl(f)p Fp(n)1744 2950 y Fo(1)1783 2936 y Fp(;)15 b(n)1878 2950 y Fo(2)1943 2936 y Fl(\025)25 b Fq(0)p Fp(;)15 b(\026)2179 2950 y Fo(1)2219 2936 y Fp(;)g(\026)2314 2950 y Fo(2)2379 2936 y Fl(\025)25 b Fq(0)g(:)h(\()p Fp(\026)2686 2950 y Fo(1)2726 2936 y Fp(=q)2812 2950 y Fo(1)2851 2936 y Fq(\))9 b(+)g Fp(n)3030 2950 y Fo(1)3079 2936 y Fl(\000)g Fp(l)3188 2903 y Fo(1)3186 2961 y Fn(j)3253 2936 y Fl(2)25 b Fe(N)3405 2950 y Fo(0)3444 2936 y Fp(;)15 b Fq(\()p Fp(\026)3574 2950 y Fo(2)3614 2936 y Fp(=q)3700 2950 y Fo(2)3739 2936 y Fq(\))9 b(+)75 3058 y Fp(n)130 3072 y Fo(2)189 3058 y Fl(\000)20 b Fp(l)309 3025 y Fo(2)307 3084 y Fn(j)374 3058 y Fl(2)25 b Fe(N)526 3072 y Fo(0)565 3058 y Fl(g)p Fq(.)216 3220 y(W)-8 b(e)32 b(ma)m(y)f(expand)118 3487 y(\010\()p Fp(w)284 3501 y Fo(1)343 3487 y Fl(\000)20 b Fp(u)486 3501 y Fo(1)526 3487 y Fp(;)15 b(w)631 3501 y Fo(2)691 3487 y Fl(\000)20 b Fp(u)834 3501 y Fo(2)873 3487 y Fq(\))p Fp(g)951 3501 y Fn(j)988 3487 y Fq(\()p Fp(u)1075 3501 y Fo(1)1115 3487 y Fp(;)15 b(u)1207 3501 y Fo(2)1247 3487 y Fq(\))26 b(=)1462 3374 y Fd(1)1432 3401 y Fh(X)1404 3595 y Fn(m)1466 3604 y Fk(1)1501 3595 y Fo(=0)1721 3401 y Fh(X)1606 3592 y Fn(n)1649 3601 y Fk(1)1683 3592 y Fo(+)p Fn(\026)1780 3601 y Fk(1)1815 3592 y Fo(=)p Fn(m)1932 3601 y Fk(1)1967 3487 y Fq(\()p Fp(w)2067 3501 y Fo(1)2127 3487 y Fl(\000)20 b Fp(u)2270 3501 y Fo(1)2310 3487 y Fq(\))2345 3450 y Fn(n)2388 3459 y Fk(1)2427 3487 y Fp(u)2479 3439 y Fn(\026)2521 3448 y Fk(1)2556 3439 y Fn(=q)2623 3448 y Fk(1)2479 3513 y Fo(1)2900 3426 y Fq(1)p 2671 3466 504 4 v 2671 3550 a(\000\(1)h(+)f Fp(\026)2975 3564 y Fo(1)3014 3550 y Fp(=q)3100 3564 y Fo(1)3140 3550 y Fq(\))3185 3487 y(\010)3251 3501 y Fn(n)3294 3510 y Fk(1)3332 3487 y Fp(g)3375 3501 y Fn(j;\026)3466 3510 y Fk(1)3505 3487 y Fq(\()p Fp(w)3605 3501 y Fo(2)3645 3487 y Fp(;)15 b(u)3737 3501 y Fo(2)3777 3487 y Fq(\))75 3776 y(where)606 3943 y(\010)672 3957 y Fn(n)715 3966 y Fk(1)753 3943 y Fp(g)796 3957 y Fn(j;\026)887 3966 y Fk(1)926 3943 y Fq(\()p Fp(w)1026 3957 y Fo(2)1066 3943 y Fp(;)g(u)1158 3957 y Fo(2)1198 3943 y Fq(\))25 b(=)g(\()1438 3829 y Fd(1)1408 3857 y Fh(X)1389 4051 y Fn(n)1432 4060 y Fk(2)1467 4051 y Fo(=0)1572 3943 y Fq(\010)1638 3957 y Fn(n)1681 3966 y Fk(1)1715 3957 y Fn(n)1758 3966 y Fk(2)1797 3943 y Fq(\()p Fp(w)1897 3957 y Fo(2)1957 3943 y Fl(\000)20 b Fp(u)2100 3957 y Fo(2)2139 3943 y Fq(\))2174 3906 y Fn(n)2217 3915 y Fk(2)2256 3943 y Fq(\)\()2375 3829 y Fd(1)2345 3857 y Fh(X)2326 4051 y Fn(\026)2368 4060 y Fk(2)2404 4051 y Fo(=0)2509 3943 y Fp(c)2548 3957 y Fn(j)t(\026)2623 3966 y Fk(1)2658 3957 y Fn(\026)2700 3966 y Fk(2)2884 3882 y Fp(u)2936 3834 y Fn(\026)2978 3843 y Fk(2)3013 3834 y Fn(=q)3080 3843 y Fk(2)2936 3908 y Fo(2)p 2749 3922 V 2749 4005 a Fq(\000\(1)h(+)f Fp(\026)3053 4019 y Fo(2)3092 4005 y Fp(=q)3178 4019 y Fo(2)3218 4005 y Fq(\))3263 3943 y(\))p Fp(:)75 4203 y Fq(On)30 b(the)g(other)h(hand,)e(it)i(is)g(w)m (ell)g(kno)m(wn)f(that)h(\(see)g([3])g(p.)40 b(228\),)993 4340 y Fh(Z)1084 4366 y Fo(1)1043 4546 y(0)1123 4463 y Fq(\(1)21 b Fl(\000)f Fp(u)p Fq(\))1402 4426 y Fn(n)1445 4435 y Fk(1)1484 4463 y Fp(u)1536 4426 y Fn(\026)1578 4435 y Fk(1)1613 4426 y Fn(=q)1680 4435 y Fk(1)1719 4463 y Fp(du)25 b Fq(=)1949 4402 y(\000\(1)c(+)f Fp(n)2253 4416 y Fo(1)2292 4402 y Fq(\)\000\(1)h(+)f(\()p Fp(\026)2666 4416 y Fo(1)2705 4402 y Fp(=q)2791 4416 y Fo(1)2831 4402 y Fq(\)\))p 1949 4443 953 4 v 2036 4526 a(\000\(2)g(+)g Fp(n)2339 4540 y Fo(1)2398 4526 y Fq(+)g(\()p Fp(\026)2579 4540 y Fo(1)2619 4526 y Fp(=q)2705 4540 y Fo(1)2744 4526 y Fq(\)\))2912 4463 y Fp(:)75 4710 y Fq(Making)31 b(the)g(c)m(hange)g (of)g(v)-5 b(ariable)31 b Fp(uw)1422 4724 y Fo(1)1487 4710 y Fq(=)25 b Fp(u)1635 4724 y Fo(1)1674 4710 y Fq(,)31 b(w)m(e)g(get)635 4831 y Fh(Z)726 4857 y Fn(w)777 4866 y Fk(1)686 5037 y Fo(0)815 4955 y Fq(\()p Fp(w)915 4969 y Fo(1)975 4955 y Fl(\000)20 b Fp(u)1118 4969 y Fo(1)1158 4955 y Fq(\))1193 4917 y Fn(n)1236 4926 y Fk(1)1275 4955 y Fp(u)1327 4907 y Fn(\026)1369 4916 y Fk(1)1404 4907 y Fn(=q)1471 4916 y Fk(1)1327 4981 y Fo(1)1509 4955 y Fp(du)1608 4969 y Fo(1)1673 4955 y Fq(=)25 b Fp(w)1836 4907 y Fn(n)1879 4916 y Fk(1)1914 4907 y Fo(+\()p Fn(\026)2038 4916 y Fk(1)2074 4907 y Fn(=q)2141 4916 y Fk(1)2175 4907 y Fo(\)+1)1834 4981 y(1)2307 4893 y Fq(\000\(1)c(+)e Fp(n)2610 4907 y Fo(1)2649 4893 y Fq(\)\000\(1)j(+)e(\()p Fp(\026)3024 4907 y Fo(1)3063 4893 y Fp(=q)3149 4907 y Fo(1)3189 4893 y Fq(\)\))p 2307 4934 V 2393 5017 a(\000\(2)h(+)f Fp(n)2697 5031 y Fo(1)2756 5017 y Fq(+)g(\()p Fp(\026)2937 5031 y Fo(1)2977 5017 y Fp(=q)3063 5031 y Fo(1)3102 5017 y Fq(\)\))3269 4955 y Fp(:)75 5201 y Fq(W)-8 b(e)32 b(deduce)e(that)75 5565 y(\(\010)20 b Fl(\003)h Fp(g)305 5579 y Fn(j)342 5565 y Fq(\))377 5579 y Fo(1)1338 5301 y Fh(Z)1429 5327 y Fn(w)1480 5336 y Fk(1)1388 5507 y Fo(0)1533 5425 y Fq(\010\()p Fp(w)1699 5439 y Fo(1)1759 5425 y Fl(\000)f Fp(u)1902 5439 y Fo(1)1942 5425 y Fp(;)15 b(w)2047 5439 y Fo(2)2107 5425 y Fl(\000)20 b Fp(u)2250 5439 y Fo(2)2289 5425 y Fq(\))p Fp(g)2367 5439 y Fn(j)2404 5425 y Fq(\()p Fp(u)2491 5439 y Fo(1)2531 5425 y Fp(;)15 b(u)2623 5439 y Fo(2)2663 5425 y Fq(\))p Fp(du)2797 5439 y Fo(1)2862 5425 y Fq(=)915 5545 y Fd(1)885 5572 y Fh(X)857 5767 y Fn(m)919 5776 y Fk(1)954 5767 y Fo(=0)1174 5572 y Fh(X)1059 5764 y Fn(n)1102 5773 y Fk(1)1136 5764 y Fo(+)p Fn(\026)1233 5773 y Fk(1)1268 5764 y Fo(=)p Fn(m)1385 5773 y Fk(1)1435 5659 y Fp(w)1502 5611 y Fn(n)1545 5620 y Fk(1)1580 5611 y Fo(+\()p Fn(\026)1704 5620 y Fk(1)1740 5611 y Fn(=q)1807 5620 y Fk(1)1841 5611 y Fo(\)+1)1500 5685 y(1)2173 5597 y Fq(\000\(1)21 b(+)f Fp(n)2477 5611 y Fo(1)2516 5597 y Fq(\))p 1973 5638 780 4 v 1973 5721 a(\000\(2)h(+)e Fp(n)2276 5735 y Fo(1)2336 5721 y Fq(+)h(\()p Fp(\026)2517 5735 y Fo(1)2556 5721 y Fp(=q)2642 5735 y Fo(1)2682 5721 y Fq(\)\))2762 5659 y(\010)2828 5673 y Fn(n)2871 5682 y Fk(1)2910 5659 y Fp(g)2953 5673 y Fn(j;\026)3044 5682 y Fk(1)3082 5659 y Fq(\()p Fp(w)3182 5673 y Fo(2)3222 5659 y Fp(;)15 b(u)3314 5673 y Fo(2)3354 5659 y Fq(\))p Fp(:)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fq(9)75 399 y(Using)31 b(the)f(same)h(calculations,)i(one)d(can)h(also)g(sho)m(w)g(that)75 769 y(\(\010)20 b Fl(\003)h Fp(g)305 783 y Fn(j)342 769 y Fq(\))377 783 y Fo(2)1607 505 y Fh(Z)1698 532 y Fn(w)1749 541 y Fk(2)1657 711 y Fo(0)1802 629 y Fq(\010)1868 643 y Fn(n)1911 652 y Fk(1)1950 629 y Fp(g)1993 643 y Fn(j;\026)2084 652 y Fk(1)2122 629 y Fq(\()p Fp(w)2222 643 y Fo(2)2262 629 y Fp(;)15 b(u)2354 643 y Fo(2)2394 629 y Fq(\))p Fp(du)2528 643 y Fo(2)2593 629 y Fq(=)1001 749 y Fd(1)971 776 y Fh(X)943 971 y Fn(m)1005 980 y Fk(2)1040 971 y Fo(=0)1260 776 y Fh(X)1145 968 y Fn(n)1188 977 y Fk(2)1223 968 y Fo(+)p Fn(\026)1320 977 y Fk(2)1354 968 y Fo(=)p Fn(m)1471 977 y Fk(2)1521 863 y Fp(w)1588 815 y Fn(n)1631 824 y Fk(2)1666 815 y Fo(+\()p Fn(\026)1790 824 y Fk(2)1826 815 y Fn(=q)1893 824 y Fk(2)1927 815 y Fo(\)+1)1586 889 y(2)2260 801 y Fq(\000\(1)20 b(+)g Fp(n)2563 815 y Fo(2)2602 801 y Fq(\))p 2059 842 780 4 v 2059 925 a(\000\(2)h(+)f Fp(n)2363 939 y Fo(2)2422 925 y Fq(+)g(\()p Fp(\026)2603 939 y Fo(2)2642 925 y Fp(=q)2728 939 y Fo(2)2768 925 y Fq(\)\))2848 863 y(\010)2914 877 y Fn(n)2957 886 y Fk(1)2992 877 y Fn(n)3035 886 y Fk(2)3073 863 y Fp(c)3112 877 y Fn(j)t(\026)3187 886 y Fk(1)3222 877 y Fn(\026)3264 886 y Fk(2)3303 863 y Fp(:)216 1150 y Fq(On)30 b(the)g(other)h(hand,)e (b)m(y)i(the)f(c)m(hange)i(of)e(v)-5 b(ariable)31 b Fp(v)2089 1105 y Fn(q)2121 1117 y Fg(k)2086 1179 y Fn(k)2189 1150 y Fq(=)25 b Fp(w)2350 1165 y Fn(k)2392 1150 y Fq(,)31 b Fp(k)d Fq(=)d(1)p Fp(;)15 b Fq(2,)32 b(w)m(e)f(get)716 1389 y(1)p 631 1430 216 4 v 631 1513 a(2)p Fp(\031)s(iq)803 1528 y Fn(k)871 1327 y Fh(I)922 1533 y Fn(q)954 1545 y Fg(k)991 1533 y Fn(C)1041 1541 y Fg(\032)1073 1559 y(k)1145 1389 y Fp(w)1212 1356 y Fn(n)1255 1368 y Fg(k)1294 1356 y Fo(+\()p Fn(\026)1418 1368 y Fg(k)1456 1356 y Fn(=q)1523 1368 y Fg(k)1561 1356 y Fo(\)+1)p 1145 1430 539 4 v 1150 1540 a Fq(\()p Fp(w)1250 1555 y Fn(k)1313 1540 y Fl(\000)20 b Fp(z)1446 1555 y Fn(k)1489 1540 y Fq(\))1524 1498 y Fo(2+)p Fn(l)1636 1476 y Fg(k)1635 1521 y(j)1693 1451 y Fp(dw)1805 1466 y Fn(k)1873 1451 y Fq(=)2023 1389 y(1)p 1979 1430 132 4 v 1979 1513 a(2)p Fp(\031)s(i)2136 1327 y Fh(I)2187 1533 y Fn(C)2237 1582 y Fg(\032)2269 1548 y Fk(1)p Fg(=q)2359 1565 y(k)2269 1610 y(k)2489 1386 y Fp(v)2536 1324 y Fn(q)2568 1336 y Fg(k)2605 1324 y Fo(\()p Fn(n)2675 1336 y Fg(k)2714 1324 y Fd(\000)p Fn(l)2791 1300 y Fg(k)2790 1345 y(j)2829 1324 y Fo(\)+)p Fn(\026)2953 1336 y Fg(k)2992 1324 y Fd(\000)p Fo(1)2533 1416 y Fn(k)p 2435 1430 704 4 v 2435 1540 a Fq(\(1)h Fl(\000)f Fq(\()p Fp(z)2704 1555 y Fn(k)2748 1540 y Fp(=v)2840 1496 y Fn(q)2872 1508 y Fg(k)2837 1569 y Fn(k)2914 1540 y Fq(\)\))2984 1498 y Fo(2+)p Fn(l)3096 1476 y Fg(k)3095 1521 y(j)3149 1451 y Fp(dv)3240 1466 y Fn(k)3283 1451 y Fp(:)216 1759 y Fq(F)-8 b(rom)31 b(the)g(expansion)f (with)g(resp)s(ect)g(to)h Fp(v)1707 1774 y Fn(k)1750 1759 y Fq(,)508 2017 y(\(1)21 b(+)f(\()p Fp(z)777 2032 y Fn(k)821 2017 y Fp(=v)913 1973 y Fn(q)945 1985 y Fg(k)910 2046 y Fn(k)987 2017 y Fq(\)\))1057 1976 y Fd(\000)p Fo(2)p Fd(\000)p Fn(l)1224 1952 y Fg(k)1223 1997 y(j)1292 2017 y Fq(=)25 b(1)c(+)1576 1903 y Fd(1)1546 1930 y Fh(X)1545 2125 y Fn(n)p Fo(=1)1703 1943 y Fq(\()p Fl(\000)p Fq(2)g Fl(\000)f Fp(l)1995 1910 y Fn(k)1993 1968 y(j)2037 1943 y Fq(\))15 b Fl(\001)g(\001)g(\001)i Fq(\()p Fl(\000)p Fq(2)k Fl(\000)e Fp(l)2500 1910 y Fn(k)2498 1968 y(j)2563 1943 y Fl(\000)h Fq(\()p Fp(n)g Fl(\000)g Fq(1\)\))p 1703 1996 1269 4 v 2297 2079 a Fp(n)p Fq(!)2981 2017 y(\()p Fl(\000)p Fp(z)3129 2032 y Fn(k)3172 2017 y Fp(=v)3264 1973 y Fn(q)3296 1985 y Fg(k)3261 2046 y Fn(k)3339 2017 y Fq(\))3374 1979 y Fn(n)75 2284 y Fq(and)30 b(the)g(residues)g (theorem,)h(w)m(e)g(deduce)f(that)75 2532 y(\()p Fp(Q)p Fq(\()p Fp(z)259 2547 y Fn(k)303 2532 y Fq(\)\))577 2471 y(1)p 493 2512 216 4 v 493 2595 a(2)p Fp(\031)s(iq)665 2610 y Fn(k)733 2409 y Fh(I)783 2615 y Fn(q)815 2627 y Fg(k)853 2615 y Fn(C)903 2623 y Fg(\032)935 2641 y(k)1007 2471 y Fp(w)1074 2438 y Fn(n)1117 2450 y Fg(k)1155 2438 y Fo(+\()p Fn(\026)1279 2450 y Fg(k)1318 2438 y Fn(=q)1385 2450 y Fg(k)1423 2438 y Fo(\)+1)p 1007 2512 539 4 v 1012 2621 a Fq(\()p Fp(w)1112 2636 y Fn(k)1175 2621 y Fl(\000)19 b Fp(z)1307 2636 y Fn(k)1350 2621 y Fq(\))1385 2580 y Fo(2+)p Fn(l)1497 2557 y Fg(k)1496 2602 y(j)1555 2532 y Fp(dw)1667 2547 y Fn(k)1735 2532 y Fq(=)2110 2471 y(\000\(2)i(+)f Fp(n)2414 2486 y Fn(k)2476 2471 y Fq(+)g(\()p Fp(\026)2657 2486 y Fn(k)2700 2471 y Fp(=q)2786 2486 y Fn(k)2829 2471 y Fq(\)\))p 1841 2512 1328 4 v 1841 2600 a(\000\(2)h(+)f Fp(l)2119 2569 y Fn(k)2117 2627 y(j)2161 2600 y Fq(\)\000\(1)h(+)f(\()p Fp(\026)2535 2615 y Fn(k)2578 2600 y Fp(=q)2664 2615 y Fn(k)2707 2600 y Fq(\))g(+)g Fp(n)2908 2615 y Fn(k)2971 2600 y Fl(\000)g Fp(l)3091 2569 y Fn(k)3089 2627 y(j)3133 2600 y Fq(\))3178 2532 y Fp(z)3224 2470 y Fo(\()p Fn(\026)3293 2482 y Fg(k)3333 2470 y Fn(=q)3400 2482 y Fg(k)3438 2470 y Fo(\)+)p Fn(n)3563 2482 y Fg(k)3601 2470 y Fd(\000)p Fn(l)3678 2446 y Fg(k)3677 2491 y(j)3220 2562 y Fn(k)3720 2532 y Fp(;)75 2810 y Fq(when)29 b(\()p Fp(\026)402 2825 y Fn(k)445 2810 y Fp(=q)531 2825 y Fn(k)574 2810 y Fq(\))20 b(+)g Fp(n)775 2825 y Fn(k)838 2810 y Fl(\000)g Fp(l)958 2777 y Fn(k)956 2835 y(j)1026 2810 y Fl(2)k Fe(N)1177 2824 y Fo(0)1216 2810 y Fq(.)41 b(The)30 b(latter)i(in)m(tegral)g(is)e (equal)h(to)g(zero)g(if)g(\()p Fp(\026)2856 2825 y Fn(k)2898 2810 y Fp(=q)2984 2825 y Fn(k)3027 2810 y Fq(\))21 b(+)f Fp(n)3229 2825 y Fn(k)3291 2810 y Fl(\000)g Fp(l)3411 2777 y Fn(k)3409 2835 y(j)3489 2810 y Fp(=)-55 b Fl(2)25 b Fe(N)3631 2824 y Fo(0)3670 2810 y Fq(.)216 2968 y(F)-8 b(rom)31 b(the)g(form)m(ula)f(\(\010)20 b Fl(\003)h Fp(g)1175 2982 y Fn(j)1212 2968 y Fq(\))1247 2982 y Fo(1)1287 2968 y Fq(,)30 b(\(\010)20 b Fl(\003)h Fp(g)1572 2982 y Fn(j)1609 2968 y Fq(\))1644 2982 y Fo(2)1714 2968 y Fq(and)30 b(\()p Fp(Q)p Fq(\()p Fp(z)2075 2983 y Fn(k)2118 2968 y Fq(\)\),)i Fp(k)c Fq(=)d(1)p Fp(;)15 b Fq(2,)32 b(w)m(e)f(deduce)f(that)431 3157 y(\000\(2)20 b(+)g Fp(l)708 3125 y Fo(1)706 3183 y Fn(j)748 3157 y Fq(\)\000\(2)h(+)f Fp(l)1061 3125 y Fo(2)1059 3183 y Fn(j)1100 3157 y Fq(\))p 431 3210 705 4 v 582 3294 a(\(2)p Fp(\031)s(i)p Fq(\))783 3267 y Fo(2)824 3294 y Fp(q)865 3308 y Fo(1)904 3294 y Fp(q)945 3308 y Fo(2)1161 3108 y Fh(I)1211 3314 y Fn(q)1243 3323 y Fk(1)1277 3314 y Fn(C)1327 3322 y Fg(\032)1359 3337 y Fk(1)1417 3108 y Fh(I)1468 3314 y Fn(q)1500 3323 y Fk(2)1534 3314 y Fn(C)1584 3322 y Fg(\032)1616 3337 y Fk(2)1674 3231 y Fq(\010)g Fl(\003)h Fp(g)1869 3245 y Fn(j)1906 3231 y Fq(\()p Fp(w)2006 3245 y Fo(1)2045 3231 y Fp(;)15 b(w)2150 3245 y Fo(2)2190 3231 y Fq(\))2447 3170 y Fp(dw)2559 3184 y Fo(1)p 2235 3210 574 4 v 2235 3318 a Fq(\()p Fp(w)2335 3332 y Fo(1)2396 3318 y Fl(\000)20 b Fp(z)2529 3332 y Fo(1)2568 3318 y Fq(\))2603 3276 y Fo(\(2+)p Fn(l)2742 3253 y Fk(1)2741 3298 y Fg(j)2778 3276 y Fo(\))3040 3170 y Fp(dw)3152 3184 y Fo(2)p 2829 3210 V 2829 3318 a Fq(\()p Fp(w)2929 3332 y Fo(2)2989 3318 y Fl(\000)g Fp(z)3122 3332 y Fo(2)3162 3318 y Fq(\))3197 3276 y Fo(\(2+)p Fn(l)3336 3253 y Fk(2)3335 3298 y Fg(j)3371 3276 y Fo(\))3438 3231 y Fq(=)652 3407 y Fd(1)622 3434 y Fh(X)536 3628 y Fn(m)598 3637 y Fk(1)633 3628 y Fn(;m)715 3637 y Fk(2)749 3628 y Fo(=0)855 3434 y Fh(X)881 3631 y Fd(B)927 3641 y Fg(j)1001 3520 y Fp(c)1040 3534 y Fn(j)t(\026)1115 3543 y Fk(1)1150 3534 y Fn(\026)1192 3543 y Fk(2)1231 3520 y Fq(\010)1297 3534 y Fn(n)1340 3543 y Fk(1)1374 3534 y Fn(n)1417 3543 y Fk(2)1778 3459 y Fp(n)1833 3473 y Fo(1)1872 3459 y Fq(!)p Fp(n)1952 3473 y Fo(2)1991 3459 y Fq(!)p Fp(z)2062 3396 y Fo(\()p Fn(\026)2131 3405 y Fk(1)2167 3396 y Fn(=q)2234 3405 y Fk(1)2269 3396 y Fo(\)+)p Fn(n)2394 3405 y Fk(1)2429 3396 y Fd(\000)p Fn(l)2506 3373 y Fk(1)2505 3418 y Fg(j)2058 3485 y Fo(1)2544 3459 y Fp(z)2590 3396 y Fo(\()p Fn(\026)2659 3405 y Fk(2)2695 3396 y Fn(=q)2762 3405 y Fk(2)2796 3396 y Fo(\)+)p Fn(n)2921 3405 y Fk(2)2956 3396 y Fd(\000)p Fn(l)3033 3373 y Fk(2)3032 3418 y Fg(j)2586 3485 y Fo(2)p 1466 3499 1919 4 v 1466 3584 a Fq(\000\(1)h(+)e(\()p Fp(\026)1804 3598 y Fo(1)1844 3584 y Fp(=q)1930 3598 y Fo(1)1969 3584 y Fq(\))i(+)f Fp(n)2171 3598 y Fo(1)2230 3584 y Fl(\000)g Fp(l)2350 3553 y Fo(1)2348 3611 y Fn(j)2389 3584 y Fq(\)\000\(1)i(+)d(\()p Fp(\026)2763 3598 y Fo(2)2803 3584 y Fp(=q)2889 3598 y Fo(2)2929 3584 y Fq(\))h(+)g Fp(n)3130 3598 y Fo(2)3189 3584 y Fl(\000)g Fp(l)3309 3553 y Fo(2)3307 3611 y Fn(j)3348 3584 y Fq(\))75 3822 y(where)j Fl(B)391 3836 y Fn(j)452 3822 y Fq(=)i Fl(f)p Fp(n)648 3836 y Fo(1)688 3822 y Fp(;)15 b(n)783 3836 y Fo(2)822 3822 y Fp(;)g(\026)917 3836 y Fo(1)957 3822 y Fp(;)g(\026)1052 3836 y Fo(2)1117 3822 y Fq(:)25 b Fp(n)1222 3836 y Fo(1)1268 3822 y Fq(+)6 b Fp(\026)1400 3836 y Fo(1)1464 3822 y Fq(=)25 b Fp(m)1640 3836 y Fo(1)1679 3822 y Fp(;)15 b(n)1774 3836 y Fo(2)1820 3822 y Fq(+)6 b Fp(\026)1952 3836 y Fo(2)2017 3822 y Fq(=)25 b Fp(m)2193 3836 y Fo(2)2232 3822 y Fp(;)15 b Fq(\()p Fp(\026)2362 3836 y Fo(1)2402 3822 y Fp(=q)2488 3836 y Fo(1)2528 3822 y Fq(\))6 b(+)g Fp(n)2701 3836 y Fo(1)2747 3822 y Fl(\000)g Fp(l)2853 3789 y Fo(1)2851 3848 y Fn(j)2918 3822 y Fl(2)24 b Fe(N)3069 3836 y Fo(0)3108 3822 y Fp(;)15 b Fq(\()p Fp(\026)3238 3836 y Fo(2)3278 3822 y Fp(=q)3364 3836 y Fo(2)3404 3822 y Fq(\))6 b(+)g Fp(n)3577 3836 y Fo(2)3623 3822 y Fl(\000)g Fp(l)3729 3789 y Fo(2)3727 3848 y Fn(j)3794 3822 y Fl(2)75 3935 y Fe(N)141 3949 y Fo(0)180 3935 y Fl(g)p Fq(.)216 4109 y(Due)32 b(to)g(the)f(expansion)g(\()p Fp(u)1187 4012 y Fa(P)1187 4136 y Fn(j)1266 4109 y Fq(\),)h(w)m(e)f (see)h(that)g(the)f(righ)m(t)h(handside)e(of)i(the)f(latter)h(form)m (ula)g(is)f(equal)h(to)75 4222 y Fp(u)127 4236 y Fn(j)164 4222 y Fq(\()p Fp(z)241 4236 y Fo(1)281 4222 y Fp(;)15 b(z)363 4236 y Fo(2)403 4222 y Fq(\))30 b(from)g(whic)m(h)g(w)m(e)h (deduce)f(the)h(exp)s(ected)g(form)m(ula.)1512 b Fc(2)75 4509 y Fb(3.2)112 b(Analytic)38 b(con)m(tin)m(uation)g(and)g(estimates) g(of)g(auxiliary)h(series)75 4681 y Fq(In)33 b(the)i(follo)m(wing,)h(w) m(e)f(will)g(use)e(the)i(notation)g(\()p Fp(a;)15 b(b)p Fq(\))35 b(to)g(denote)g(the)f(op)s(en)g(in)m(terv)-5 b(al)35 b(\()p Fp(a;)15 b(b)p Fq(\))35 b(or)f(an)m(y)h(of)f(its)75 4794 y(translates)f(\()p Fp(a)22 b Fq(+)g(2)p Fp(k)s(\031)s(;)15 b(b)22 b Fq(+)g(2)p Fp(k)s(\031)s Fq(\),)34 b Fp(k)e Fl(2)c Fe(Z)p Fq(.)46 b(W)-8 b(e)34 b(will)f(denote)g(b)m(y)f Fp(G)2405 4809 y Fn(\016)2443 4794 y Fq(\()p Fp(\025)2531 4808 y Fo(0)2571 4794 y Fq(\))h(an)m(y)g(sector)g(of)g Fe(C)p Fq(,)f(cen)m(tered)i(at)f(0,)75 4907 y(whic)m(h)d(do)s(es)g(not) h(in)m(tersect)h Fp(D)s Fq(\()p Fp(\025)1235 4868 y Fd(\000)p Fo(1)1235 4933 y(0)1329 4907 y Fp(;)15 b(\016)s Fq(\),)1160 5098 y Fp(G)1231 5113 y Fn(\016)1269 5098 y Fq(\()p Fp(\025)1357 5112 y Fo(0)1397 5098 y Fq(\))26 b(=)f Fl(f)p Fp(t)g Fq(=)g Fl(j)p Fp(t)p Fl(j)p Fp(e)1878 5060 y Fn(i\022)1935 5068 y Fg(t)1993 5098 y Fq(:)g Fp(\022)2086 5112 y Fn(t)2141 5098 y Fl(2)g Fq(\()p Fp(d)2309 5060 y Fo(1)2309 5121 y Fn(\025)2350 5130 y Fk(0)2385 5121 y Fn(;\016)2443 5098 y Fp(;)15 b(d)2530 5060 y Fo(2)2530 5121 y Fn(\025)2571 5130 y Fk(0)2606 5121 y Fn(;\016)2664 5098 y Fq(\))p Fl(g)p Fp(:)216 5318 y Fq(In)38 b(the)h(follo)m(wing,)j(w)m(e)d(put)f Fp(q)s Fq(\()p Fp(u)1370 5332 y Fo(1)1410 5318 y Fp(;)15 b(u)1502 5332 y Fo(2)1542 5318 y Fq(\))39 b(=)g Fp(\025)p Fq(\()p Fp(u)1866 5332 y Fo(1)1906 5318 y Fp(;)15 b(u)1998 5332 y Fo(2)2038 5318 y Fq(\))p Fp(=)p Fq(\()p Fp(u)2205 5270 y Fn(l)2226 5279 y Fk(1)2262 5270 y Fn(=q)2329 5279 y Fk(1)2205 5344 y Fo(1)2367 5318 y Fp(u)2419 5270 y Fn(l)2440 5279 y Fk(2)2475 5270 y Fn(=q)2542 5279 y Fk(2)2419 5344 y Fo(2)2580 5318 y Fq(\).)66 b(F)-8 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1837 5651 1337 4 v 1837 5760 a(u)1889 5712 y Fo(1)p Fn(=q)1991 5721 y Fk(1)1889 5786 y Fo(1)2030 5760 y Fp(u)2082 5712 y Fo(1)p Fn(=q)2184 5721 y Fk(2)2082 5786 y Fo(2)2223 5760 y Fq(\(1)21 b Fl(\000)f Fp(q)s Fq(\()p Fp(u)2546 5774 y Fo(1)2586 5760 y Fp(;)15 b(u)2678 5774 y Fo(2)2717 5760 y Fq(\))p Fp(t)p Fq(\)\(1)22 b Fl(\000)e Fp(\025)3066 5774 y Fo(0)3105 5760 y Fp(t)p Fq(\))p eop end %%Page: 10 10 TeXDict begin 10 9 bop 75 100 a Fq(10)75 399 y(is)30 b(w)m(ell)i(de\014ned,)d(where)h Fp(q)996 366 y Fd(0)1019 399 y Fq(\()p Fp(u)1106 413 y Fo(1)1146 399 y Fp(;)15 b(u)1238 413 y Fo(2)1278 399 y Fq(\))31 b(is)f(b)s(ounded)e(b)m(y)i(a)h (constan)m(t)h(dep)s(ending)d(only)h(on)g Fp(R)3206 413 y Fo(1)3246 399 y Fp(;)15 b(R)3355 413 y Fo(2)3395 399 y Fq(.)216 566 y(F)-8 b(or)31 b Fp(t)25 b Fl(2)g Fp(G)594 581 y Fn(\016)633 566 y Fq(\()p Fp(\025)721 580 y Fo(0)760 566 y Fq(\))31 b(and)f Fp(z)1045 580 y Fo(1)1085 566 y Fp(;)15 b(z)1167 580 y Fo(2)1232 566 y Fl(2)25 b Fe(C)p Fq(,)30 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b(is)e(the)h(Hank)m(el)h(path)f (\(see)h([3],)h(p.)53 b(228\).)j(F)-8 b(rom)35 b(the)g(decomp)s (osition)g(\()p Fp(F)13 b Fq(\),)37 b(w)m(e)75 1201 y(deduce)30 b(that)1288 1333 y Fp(k)s Fq(\()p Fp(t;)15 b(z)1488 1347 y Fo(1)1529 1333 y Fp(;)g(z)1611 1347 y Fo(2)1651 1333 y Fq(\))25 b(=)1935 1272 y(1)p 1817 1312 282 4 v 1817 1396 a(1)c Fl(\000)f Fp(\025)2027 1410 y Fo(0)2066 1396 y Fp(t)2129 1333 y Fq(+)g Fp(B)5 b Fq(\()p Fp(t;)15 b(z)2444 1347 y Fo(1)2484 1333 y Fp(;)g(z)2566 1347 y Fo(2)2606 1333 y Fq(\))75 1540 y(where)241 1775 y Fp(B)5 b Fq(\()p Fp(t;)15 b(z)465 1789 y Fo(1)504 1775 y Fp(;)g(z)586 1789 y Fo(2)627 1775 y Fq(\))25 b(=)891 1714 y(1)p 793 1755 242 4 v 793 1838 a(\(2)p Fp(\031)s(i)p Fq(\))994 1812 y Fo(2)1060 1652 y Fh(I)1111 1858 y Fn(\015)t Fo(\()p Fn(R)1231 1867 y Fk(1)1266 1858 y Fn(;z)1319 1867 y Fk(1)1353 1858 y Fo(\))1400 1652 y Fh(I)1450 1858 y Fn(\015)t Fo(\()p Fn(R)1570 1867 y Fk(2)1606 1858 y Fn(;z)1659 1867 y Fk(2)1693 1858 y Fo(\))2176 1714 y Fp(e)2218 1681 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y Fp(;)15 b(L)692 4752 y Fn(j)692 4822 y Fo(2)732 4796 y Fq(\),)31 b(\()p Fp(L)920 4752 y Fn(j)920 4822 y Fo(1)960 4796 y Fp(;)15 b(C)1065 4810 y Fo(2)1104 4796 y Fq(\),)31 b Fp(j)g Fq(=)25 b(1)p Fp(;)15 b Fq(2.)216 4963 y(W)-8 b(e)37 b(sho)m(w)f(the)g (existence)i(and)d(estimates)j(for)e(the)g(in)m(tegral)i(o)m(v)m(er)f (\()p Fp(L)2693 4930 y Fo(1)2693 4988 y(1)2732 4963 y Fp(;)15 b(C)2837 4977 y Fo(2)2877 4963 y Fq(\),)38 b(the)e(other)h (ones)f(can)g(b)s(e)75 5076 y(treated)31 b(in)f(the)h(same)g(manner,) 369 5328 y Fl(j)409 5204 y Fh(I)460 5410 y Fn(L)508 5387 y Fk(1)508 5431 y(1)562 5204 y Fh(I)613 5410 y Fn(C)663 5419 y Fk(2)727 5266 y Fp(e)769 5233 y Fn(u)810 5242 y Fk(1)845 5233 y Fn(z)878 5242 y Fk(1)916 5266 y Fp(e)958 5233 y Fn(u)999 5242 y Fk(2)1034 5233 y Fn(z)1067 5242 y Fk(2)1112 5266 y Fq(~)-51 b Fp(q)r Fq(\()p Fp(t;)15 b(u)1309 5280 y Fo(1)1349 5266 y Fp(;)g(u)1441 5280 y Fo(2)1481 5266 y Fq(\))p 727 5307 790 4 v 784 5415 a Fp(u)836 5367 y Fo(1+\(1)p Fn(=q)1055 5376 y 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Fp(R)2630 5491 y Fo(1)p Fn(=q)2732 5500 y Fk(1)2629 5565 y Fo(1)2772 5539 y Fp(R)2842 5491 y Fo(1)p Fn(=q)2944 5500 y Fk(2)2841 5565 y Fo(2)2982 5539 y Fq(\)\))p Fp(e)3094 5502 y Fn(R)3147 5511 y Fk(1)3183 5502 y Fd(j)p Fn(z)3236 5511 y Fk(1)3270 5502 y Fd(j)3294 5539 y Fp(e)3336 5502 y Fn(R)3389 5511 y Fk(2)3424 5502 y Fd(j)p Fn(z)3477 5511 y Fk(2)3511 5502 y Fd(j)3535 5539 y Fp(;)75 5740 y Fq(from)30 b(whic)m(h)g(the)h (estimate)h(of)e(the)h(prop)s(osition)f(5)h(follo)m(ws.)1579 b Fc(2)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 3764 100 a Fq(11)75 399 y(In)30 b(the)g(follo)m(wing,)i(w)m(e)f(consider)f(the)h(series)1258 663 y Fp(v)s Fq(\()p Fp(t;)15 b(z)1455 677 y Fo(1)1496 663 y Fp(;)g(z)1578 677 y Fo(2)1618 663 y Fq(\))26 b(=)1805 549 y Fd(1)1775 576 y Fh(X)1779 772 y Fn(j)t Fo(=0)2067 601 y Fp(u)2119 615 y Fn(j)2156 601 y Fq(\()p Fp(z)2233 615 y Fo(1)2273 601 y Fp(;)15 b(z)2355 615 y Fo(2)2395 601 y Fq(\))p Fp(t)2463 568 y Fn(j)p 1931 642 705 4 v 1931 727 a Fq(\000\(1)21 b(+)f 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Fp(k)s Fq(.)75 1537 y Ff(Prop)s(osition)36 b(6)46 b Fm(F)-7 b(or)34 b(al)5 b(l)33 b Fp(t)p Fm(,)f(such)h(that)h Fl(j)p Fp(t)p Fl(j)25 b Fp(<)g(\032)33 b Fm(with)g Fp(\032)g Fm(su\016ciently)g(smal)5 b(l)33 b(and)h Fq(\()p Fp(z)3039 1551 y Fo(1)3079 1537 y Fp(;)15 b(z)3161 1551 y Fo(2)3201 1537 y Fq(\))25 b Fl(2)g Fp(D)3422 1551 y Fn(\032)3458 1560 y Fk(1)3518 1537 y Fl(\002)20 b Fp(D)3684 1551 y Fn(\032)3720 1560 y Fk(2)3759 1537 y Fm(,)352 1807 y Fp(v)s Fq(\()p Fp(t;)15 b(z)549 1821 y Fo(1)589 1807 y Fp(;)g(z)671 1821 y Fo(2)711 1807 y Fq(\))26 b(=)976 1746 y(1)p 878 1786 242 4 v 878 1869 a(\(2)p Fp(\031)s(i)p Fq(\))1079 1843 y Fo(2)1130 1807 y Fp(@)1178 1821 y Fn(z)1211 1830 y Fk(1)1250 1807 y Fp(@)1298 1821 y Fn(z)1331 1830 y Fk(2)1385 1683 y Fh(I)1435 1889 y Fn(C)1485 1936 y Fg(\032)1517 1904 y Fk(1)p Fg(=q)1607 1919 y Fk(1)1517 1961 y(1)1670 1683 y Fh(I)1721 1889 y Fn(C)1771 1936 y Fg(\032)1803 1904 y Fk(1)p Fg(=q)1893 1919 y Fk(2)1803 1961 y(2)1956 1807 y Fq(\010)20 b Fl(\003)g Fp(k)s Fq(\()p Fp(t;)15 b(v)2309 1821 y Fo(1)2350 1807 y Fp(;)g(v)2434 1821 y Fo(2)2473 1807 y Fq(\))2692 1746 y Fp(v)2739 1701 y Fn(q)2771 1710 y Fk(1)2806 1701 y Fd(\000)p Fo(1)2736 1772 y(1)2900 1746 y Fp(v)2947 1701 y Fn(q)2979 1710 y Fk(2)3013 1701 y Fd(\000)p Fo(1)2944 1772 y(2)p 2518 1786 763 4 v 2518 1870 a Fq(\()p Fp(v)2600 1826 y Fn(q)2632 1835 y Fk(1)2597 1896 y Fo(1)2692 1870 y Fl(\000)20 b Fp(z)2825 1884 y Fo(1)2865 1870 y Fq(\)\()p Fp(v)2982 1826 y Fn(q)3014 1835 y Fk(2)2979 1896 y Fo(2)3073 1870 y Fl(\000)g Fp(z)3206 1884 y Fo(2)3246 1870 y Fq(\))3291 1807 y Fp(dv)3382 1821 y Fo(1)3422 1807 y Fp(dv)3513 1821 y Fo(2)3553 1807 y Fp(;)75 2119 y Fm(wher)-5 b(e)109 2371 y Fq(\010)20 b Fl(\003)g Fp(k)s Fq(\()p Fp(t;)15 b(v)462 2385 y Fo(1)503 2371 y Fp(;)g(v)587 2385 y Fo(2)627 2371 y Fq(\))25 b(=)783 2247 y Fh(Z)874 2273 y Fn(v)910 2236 y Fg(q)938 2251 y Fk(2)908 2295 y(2)834 2453 y Fo(0)997 2247 y Fh(Z)1088 2273 y Fn(v)1124 2236 y Fg(q)1152 2251 y Fk(1)1122 2295 y(1)1048 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4290 a(2)p Fp(\031)s(i)1969 4104 y Fh(I)2019 4310 y Fn(\015)t Fo(\()p Fn(R)2139 4322 y Fg(l)2164 4310 y Fn(;z)2217 4322 y Fg(l)2241 4310 y Fo(\))2392 4166 y Fp(e)2434 4133 y Fn(u)2475 4145 y Fg(l)2500 4133 y Fn(z)2533 4145 y Fg(l)p 2297 4207 359 4 v 2297 4316 a Fp(u)2349 4268 y Fo(1+\()p Fn(\026)2508 4280 y Fg(l)2534 4268 y Fn(=q)2601 4280 y Fg(l)2625 4268 y Fo(\))2349 4345 y Fn(l)2666 4228 y Fp(du)2765 4243 y Fn(l)2791 4228 y Fp(;)75 4523 y Fq(w)m(e)32 b(deduce)e(the)i(expansion)f(\()p Fp(k)1179 4440 y Fa(P)1258 4523 y Fq(\).)43 b(Finally)-8 b(,)33 b(from)e(the)g(in)m(tegral)i(represen)m(tation)f(of)f Fp(u)3145 4537 y Fn(j)3182 4523 y Fq(\()p Fp(z)3259 4537 y Fo(1)3299 4523 y Fp(;)15 b(z)3381 4537 y Fo(2)3421 4523 y Fq(\))31 b(in)g(prop)s(o-)75 4636 y(sition)d(4,)g(where)f(w)m(e) h(mak)m(e)g(the)f(c)m(hange)i(of)e(v)-5 b(ariable)28 b Fp(w)1997 4651 y Fn(l)2048 4636 y Fq(=)d Fp(v)2191 4592 y Fn(q)2223 4604 y Fg(l)2188 4665 y Fn(l)2252 4636 y Fq(,)j Fp(l)f Fq(=)e(1)p Fp(;)15 b Fq(2,)29 b(and)e(the)g(latter)i (expansion)e(\()p Fp(k)3715 4553 y Fa(P)3794 4636 y Fq(\),)75 4749 y(w)m(e)k(deduce)f(the)g(exp)s(ected)h(in)m(tegral)h(form)m(ula.) 2050 b Fc(2)216 5007 y Fq(In)22 b(the)h(follo)m(wing,)j(w)m(e)d(will)f (try)h(to)g(con)m(tin)m(ue)h(analytically)h(and)d(giv)m(e)i(gro)m(wth)f (estimates)h(of)f(the)g(function)75 5120 y Fp(v)s Fq(\()p Fp(t;)15 b(z)272 5134 y Fo(1)313 5120 y Fp(;)g(z)395 5134 y Fo(2)435 5120 y Fq(\))31 b(with)f(resp)s(ect)g(to)h Fp(t)g Fq(in)f(a)h(sector)g Fp(S)1695 5135 y Fn(\016)1758 5120 y Fl(\032)25 b Fe(C)p Fq(.)41 b(W)-8 b(e)32 b(will)f(giv)m(e)h (simple)e(su\016cien)m(t)h(conditions)g(on)f(the)75 5233 y(function)j(\010)h(in)f(suc)m(h)g(a)h(w)m(a)m(y)h(that)f(the)g(follo)m (wing)h(prop)s(erties)f(are)g(satis\014ed.)50 b(There)34 b(exists)g(a)g(curv)m(e)g Fl(C)39 b Fq(in)75 5345 y Fe(C)141 5312 y Fo(2)210 5345 y Fq(whic)m(h)30 b(is)g(homotopic)h(to)g(the)f (torus)g Fp(C)1570 5389 y Fn(\032)1606 5351 y Fk(1)p Fg(=q)1696 5366 y Fk(1)1606 5411 y(1)1759 5345 y Fl(\002)19 b Fp(C)1914 5389 y Fn(\032)1950 5351 y Fk(1)p Fg(=q)2040 5366 y Fk(2)1950 5411 y(2)2114 5345 y Fq(suc)m(h)30 b(that)h(the)f (function)g(\010)19 b Fl(\003)h Fp(k)s Fq(\()p Fp(t;)15 b(v)3380 5359 y Fo(1)3420 5345 y Fp(;)g(v)3504 5359 y Fo(2)3544 5345 y Fq(\))30 b(is)g(w)m(ell)75 5502 y(de\014ned)k(for)g Fp(t)f Fl(2)g Fp(S)752 5517 y Fn(\016)789 5502 y Fq(,)j(\()p Fp(v)929 5516 y Fo(1)969 5502 y Fp(;)15 b(v)1053 5516 y Fo(2)1093 5502 y Fq(\))33 b Fl(2)g(C)5 b Fq(.)54 b(Moreo)m(v)m(er,)39 b(there)c(exist)h(constan)m(ts)2702 5480 y(~)2682 5502 y Fp(C)j(>)33 b Fq(0)i(and)3176 5480 y(~)3151 5502 y Fp(K)40 b(>)33 b Fq(0,)j(suc)m(h)f(that)75 5615 y(\010)20 b Fl(\003)h Fp(k)33 b Fq(has)d(gro)m(wth)h(estimates)1437 5740 y Fl(j)p Fq(\010)20 b Fl(\003)h Fp(k)s Fq(\()p Fp(t;)15 b(v)1816 5754 y Fo(1)1856 5740 y Fp(;)g(v)1940 5754 y Fo(2)1980 5740 y Fq(\))p Fl(j)26 b(\024)2182 5717 y Fq(~)2162 5740 y Fp(C)6 b(e)2294 5686 y Fo(~)2275 5703 y Fn(K)f Fd(j)p Fn(t)p Fd(j)2405 5679 y Fg(m)2467 5740 y Fp(;)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 75 100 a Fq(12)75 399 y(for)28 b Fp(t)d Fl(2)g Fp(S)412 414 y Fn(\016)450 399 y Fq(,)k(\()p Fp(v)583 413 y Fo(1)623 399 y Fp(;)15 b(v)707 413 y Fo(2)747 399 y Fq(\))25 b Fl(2)g(C)5 b Fq(,)29 b(where)f Fp(m)d Fq(=)g(\(\()p Fp(l)1559 413 y Fo(1)1599 399 y Fp(=q)1685 413 y Fo(1)1725 399 y Fq(\))17 b(+)f(\()p Fp(l)1926 413 y Fo(2)1966 399 y Fp(=q)2052 413 y Fo(2)2091 399 y Fq(\))h Fl(\000)f Fq(1\))2310 366 y Fd(\000)p Fo(1)2405 399 y Fq(.)40 b(Simple)28 b(calculations)i(then)f(sho)m(w)f(that)75 511 y Fp(v)s Fq(\()p Fp(t;)15 b(z)272 525 y Fo(1)313 511 y Fp(;)g(z)395 525 y Fo(2)435 511 y Fq(\))30 b(has)h(an)f(analytic) i(con)m(tin)m(uation)g(and)e(satis\014es)h(the)f(same)h(gro)m(wth)g (estimates)h(as)e(\010)20 b Fl(\003)h Fp(k)s Fq(.)216 626 y(No)m(w,)33 b(consider)f(a)h(sector)g Fp(S)1205 641 y Fn(\016)1270 626 y Fq(=)28 b Fl(f)p Fp(t)g Fq(=)g Fl(j)p Fp(t)p Fl(j)p Fp(e)1699 593 y Fn(i\022)1756 601 y Fg(t)1816 626 y Fq(:)g Fp(\022)1912 640 y Fn(t)1970 626 y Fl(2)f Fq(\()p Fp(D)2168 641 y Fo(1)p Fn(;\016)2261 626 y Fp(;)15 b(D)2376 641 y Fo(2)p Fn(;\016)2470 626 y Fq(\))p Fl(g)p Fq(.)46 b(W)-8 b(e)33 b(denote)g(b)m(y)f Fl(M)g Fq(the)g(subset)f(of)75 739 y Fe(C)141 706 y Fo(2)180 739 y Fq(,)284 984 y Fl(M)26 b Fq(=)f Fl(f)p Fq(\()p Fp(v)639 998 y Fo(1)679 984 y Fp(;)15 b(v)763 998 y Fo(2)803 984 y Fq(\))25 b Fl(2)g Fe(C)1015 947 y Fo(2)1080 984 y Fq(:)g(arg)r(\()p Fp(v)1340 940 y Fn(q)1372 949 y Fk(1)1337 1010 y Fo(1)1431 984 y Fl(\000)20 b Fp(z)1564 998 y Fo(1)1604 984 y Fq(\))1653 897 y Fg(l)1673 912 y Fk(1)p 1649 918 64 3 v 1649 959 a Fg(q)1677 974 y Fk(1)1746 984 y Fq(+)g(arg)r(\()p Fp(v)2047 940 y Fn(q)2079 949 y Fk(2)2044 1010 y Fo(2)2138 984 y Fl(\000)g Fp(z)2271 998 y Fo(2)2311 984 y Fq(\))2360 897 y Fg(l)2380 912 y Fk(2)p 2356 918 V 2356 959 a Fg(q)2384 974 y Fk(2)2458 984 y Fl(2)25 b Fq(\()p Fp(D)2654 999 y Fo(2)p Fn(;\016)2768 984 y Fl(\000)20 b Fp(d)2906 947 y Fo(2)2906 1007 y Fn(\025)2947 1016 y Fk(0)2982 1007 y Fn(;\016)3039 984 y Fp(;)15 b(D)3154 999 y Fo(1)p Fn(;\016)3268 984 y Fl(\000)20 b Fp(d)3406 947 y Fo(1)3406 1007 y Fn(\025)3447 1016 y Fk(0)3482 1007 y Fn(;\016)3539 984 y Fq(\))p Fl(g)p Fp(:)75 1195 y Fq(Let)33 b([)p Fp(m)345 1209 y Fo(1)p Fn(;j)436 1195 y Fp(;)15 b(m)556 1209 y Fo(1)p Fn(;j)t Fo(+1)738 1195 y Fq(],)34 b([)p Fp(m)927 1210 y Fo(2)p Fn(;k)1024 1195 y Fp(;)15 b(m)1144 1210 y Fo(2)p Fn(;k)r Fo(+1)1332 1195 y Fq(],)34 b(1)29 b Fl(\024)f Fp(j)35 b Fl(\024)28 b Fp(\013)1818 1209 y Fo(1)1858 1195 y Fq(,)33 b(1)c Fl(\024)g Fp(k)j Fl(\024)c Fp(\013)2326 1209 y Fo(2)2398 1195 y Fq(b)s(e)k(in)m(terv)-5 b(als)33 b(included)f(in)g(\() p Fl(\000)p Fp(\031)s(;)15 b(\031)s Fq(\).)48 b(W)-8 b(e)75 1308 y(denote)31 b(b)m(y)f Fl(M)603 1322 y Fo(1)643 1308 y Fq(,)g(the)h(subset)f(of)g Fe(C)p Fq(,)1052 1518 y Fl(M)1161 1532 y Fo(1)1226 1518 y Fq(=)25 b Fl(f)p Fp(v)1411 1532 y Fo(1)1476 1518 y Fl(2)g Fe(C)g Fq(:)g(arg)r(\()p Fp(v)1910 1532 y Fo(1)1950 1518 y Fq(\))g Fl(2)g([)2157 1477 y Fn(\013)2202 1486 y Fk(1)2157 1545 y Fn(j)t Fo(=1)2283 1518 y Fq([)p Fp(m)2388 1532 y Fo(1)p Fn(;j)2480 1518 y Fp(;)15 b(m)2600 1532 y Fo(1)p Fn(;j)t Fo(+1)2782 1518 y Fq(])p Fl(g)p Fp(;)75 1728 y Fq(and)30 b(lik)m(ewise)i(w)m(e)e (denote)h(b)m(y)g Fl(M)1248 1742 y Fo(2)1287 1728 y Fq(,)g(the)f (subset)g(of)h Fe(C)p Fq(,)1043 1939 y Fl(M)1152 1953 y Fo(2)1216 1939 y Fq(=)25 b Fl(f)p Fp(v)1401 1953 y Fo(2)1467 1939 y Fl(2)f Fe(C)h Fq(:)h(arg)r(\()p Fp(v)1901 1953 y Fo(2)1941 1939 y Fq(\))f Fl(2)g([)2148 1898 y Fn(\013)2193 1907 y Fk(2)2148 1968 y Fn(k)r Fo(=1)2280 1939 y Fq([)p Fp(m)2385 1954 y Fo(2)p Fn(;k)2483 1939 y Fp(;)15 b(m)2603 1954 y Fo(2)p Fn(;k)r Fo(+1)2791 1939 y Fq(])p Fl(g)p Fp(:)75 2149 y Fq(W)-8 b(e)24 b(assume)e(that)h(the)g (upp)s(er)e(men)m(tioned)i(in)m(terv)-5 b(als)24 b(are)f(c)m(hosen)g (suc)m(h)f(that)h(the)g(follo)m(wing)h(inclusion)f(holds,)925 2360 y Fl(N)38 b Fq(=)25 b Fl(f)p Fq(\()p Fp(v)1258 2374 y Fo(1)1298 2360 y Fp(;)15 b(v)1382 2374 y Fo(2)1422 2360 y Fq(\))26 b Fl(2)f Fe(C)1635 2322 y Fo(2)1699 2360 y Fq(:)h Fp(v)1794 2374 y Fo(1)1868 2360 y Fp(=)-55 b Fl(2)25 b(M)2053 2374 y Fo(1)2153 2360 y Fq(and)60 b Fp(v)2404 2374 y Fo(2)2479 2360 y Fp(=)-55 b Fl(2)25 b(M)2664 2374 y Fo(2)2704 2360 y Fl(g)g(\032)g(M)p Fp(:)216 2572 y Fq(W)-8 b(e)30 b(mak)m(e)g(no)m(w)e(an)h(assumption)f(on)h(the)g (function)f(\010.)40 b(The)28 b(function)g(\010\()p Fp(w)2870 2528 y Fn(q)2902 2537 y Fk(1)2868 2598 y Fo(1)2941 2572 y Fp(;)15 b(w)3048 2528 y Fn(q)3080 2537 y Fk(2)3046 2598 y Fo(2)3119 2572 y Fq(\))29 b(has)g(an)f(analytical)75 2685 y(con)m(tin)m(uation)k(on)f Fl(M)836 2699 y Fo(1)895 2685 y Fl(\002)20 b(M)1095 2699 y Fo(2)1165 2685 y Fq(and)30 b(there)g(exist)i Fp(C)f(>)25 b Fq(0,)31 b Fp(K)2160 2699 y Fo(1)2225 2685 y Fp(>)25 b Fq(0,)31 b Fp(K)2499 2699 y Fo(2)2564 2685 y Fp(>)25 b Fq(0,)31 b(suc)m(h)f(that)75 2905 y(\()p Fp(C)7 b(d)p Fq(3\))937 b Fl(j)p Fq(\010\()p Fp(w)1439 2861 y Fn(q)1471 2870 y Fk(1)1437 2931 y Fo(1)1510 2905 y Fp(;)15 b(w)1617 2861 y Fn(q)1649 2870 y Fk(2)1615 2931 y Fo(2)1688 2905 y Fq(\))p Fl(j)26 b(\024)f Fp(C)7 b(e)1984 2868 y Fn(K)2044 2877 y Fk(1)2078 2868 y Fd(j)p Fn(w)2151 2830 y Fg(q)2179 2845 y Fk(1)2149 2889 y(1)2217 2868 y Fd(j)2237 2844 y Fg(k)2270 2859 y Fk(1)2313 2905 y Fp(e)2355 2868 y Fn(K)2415 2877 y Fk(2)2449 2868 y Fd(j)p Fn(w)2522 2830 y Fg(q)2550 2845 y Fk(2)2520 2889 y(2)2588 2868 y Fd(j)2608 2844 y Fg(k)2641 2859 y Fk(2)75 3116 y Fq(for)30 b(all)h(\()p Fp(w)440 3130 y Fo(1)480 3116 y Fp(;)15 b(w)585 3130 y Fo(2)625 3116 y Fq(\))26 b Fl(2)f(M)881 3130 y Fo(1)940 3116 y Fl(\002)20 b(M)1140 3130 y Fo(2)1180 3116 y Fq(,)31 b(where)e Fp(k)1545 3130 y Fo(1)1610 3116 y Fp(>)c Fq(0,)31 b Fp(k)1854 3130 y Fo(2)1919 3116 y Fp(>)25 b Fq(0)31 b(satisfy)g(the)g(follo)m(wing)g (assumption.)75 3285 y(If)f Fp(l)193 3299 y Fo(1)258 3285 y Fl(6)p Fq(=)25 b(0)30 b(and)g Fp(l)633 3299 y Fo(2)698 3285 y Fl(6)p Fq(=)25 b(0,)31 b(then)881 3528 y(sup)o(\(sup)o(\(1)p Fp(;)15 b(k)1357 3542 y Fo(1)1398 3528 y Fq(\))1443 3467 y Fp(q)1484 3481 y Fo(1)p 1443 3507 81 4 v 1450 3591 a Fp(l)1477 3605 y Fo(1)1533 3528 y Fp(;)g Fq(sup)o(\(1)p Fp(;)g(k)1877 3542 y Fo(2)1918 3528 y Fq(\))1963 3467 y Fp(q)2004 3481 y Fo(2)p 1963 3507 V 1970 3591 a Fp(l)1997 3605 y Fo(2)2054 3528 y Fq(\))25 b Fl(\024)2594 3467 y Fq(1)p 2220 3507 794 4 v 2220 3591 a(\()p Fp(l)2282 3605 y Fo(1)2322 3591 y Fp(=q)2408 3605 y Fo(1)2448 3591 y Fq(\))20 b(+)g(\()p Fp(l)2656 3605 y Fo(2)2696 3591 y Fp(=q)2782 3605 y Fo(2)2822 3591 y Fq(\))g Fl(\000)g Fq(1)3023 3528 y Fp(:)75 3795 y Fq(Notice)38 b(that)e(the)g(latter)i(inequalit)m(y)f(implies)f(that)h (sup)n(\()p Fp(q)2140 3809 y Fo(1)2180 3795 y Fp(=l)2252 3809 y Fo(1)2292 3795 y Fp(;)15 b(q)2373 3809 y Fo(2)2412 3795 y Fp(=l)2484 3809 y Fo(2)2524 3795 y Fq(\))35 b Fl(\024)f Fq(\(\()p Fp(l)2796 3809 y Fo(1)2837 3795 y Fp(=q)2923 3809 y Fo(1)2962 3795 y Fq(\))24 b(+)g(\()p Fp(l)3178 3809 y Fo(2)3218 3795 y Fp(=q)3304 3809 y Fo(2)3343 3795 y Fq(\))h Fl(\000)e Fq(1\))3577 3762 y Fd(\000)p Fo(1)3708 3795 y Fq(and)75 3908 y(therefore)31 b(that)g Fp(l)681 3922 y Fo(1)720 3908 y Fp(=q)806 3922 y Fo(1)871 3908 y Fl(\024)25 b Fq(1)31 b(and)f Fp(l)1247 3922 y Fo(2)1286 3908 y Fp(=q)1372 3922 y Fo(2)1437 3908 y Fl(\024)25 b Fq(1.)75 4077 y(If)30 b Fp(l)193 4091 y Fo(2)258 4077 y Fq(=)25 b(0)30 b(then)1421 4235 y Fp(k)1468 4249 y Fo(1)1533 4235 y Fq(=)1753 4173 y Fp(l)1780 4187 y Fo(1)1819 4173 y Fp(=q)1905 4187 y Fo(1)p 1639 4214 420 4 v 1639 4297 a Fq(\()p Fp(l)1701 4311 y Fo(1)1741 4297 y Fp(=q)1827 4311 y Fo(1)1866 4297 y Fq(\))21 b Fl(\000)f Fq(1)2129 4235 y Fp(;)76 b(k)2277 4249 y Fo(2)2342 4235 y Fp(>)25 b Fq(0)p Fp(;)75 4452 y Fq(If)30 b Fp(l)193 4466 y Fo(1)258 4452 y Fq(=)25 b(0)30 b(then)1421 4610 y Fp(k)1468 4624 y Fo(1)1533 4610 y Fp(>)25 b Fq(0)61 b Fp(;)76 b(k)1883 4624 y Fo(2)1948 4610 y Fq(=)2168 4549 y Fp(l)2195 4563 y Fo(2)2234 4549 y Fp(=q)2320 4563 y Fo(2)p 2054 4589 V 2054 4673 a Fq(\()p Fp(l)2116 4687 y Fo(2)2156 4673 y Fp(=q)2242 4687 y Fo(2)2281 4673 y Fq(\))21 b Fl(\000)f Fq(1)2483 4610 y Fp(:)75 4857 y Ff(Prop)s(osition)36 b(7)46 b Fm(Ther)-5 b(e)34 b(exists)g Fp(\017)27 b(>)g Fq(0)34 b Fm(such)g(that)g(the)g(function)g Fp(v)s Fq(\()p Fp(t;)15 b(z)2593 4871 y Fo(1)2634 4857 y Fp(;)g(z)2716 4871 y Fo(2)2756 4857 y Fq(\))34 b Fm(has)g(an)g(analytic)-5 b(al)36 b(c)-5 b(ontinu-)75 4970 y(ation)31 b(on)g(the)g(domain)h Fq(\()p Fp(S)998 4985 y Fn(\016)1051 4970 y Fl([)15 b Fp(D)s Fq(\(0)p Fp(;)g(\032)p Fq(\)\))g Fl(\002)g Fp(D)s Fq(\(0)p Fp(;)g(\017)p Fq(\))g Fl(\002)g Fp(D)s Fq(\(0)p Fp(;)g(\017)p Fq(\))p Fm(,)34 b(and)d(ther)-5 b(e)32 b(exists)2912 4947 y Fq(~)2891 4970 y Fp(C)g(>)25 b Fq(0)p Fm(,)3212 4947 y Fq(~)3188 4970 y Fp(K)32 b(>)25 b Fq(0)p Fm(,)31 b(such)f(that)75 5083 y(for)j(al)5 b(l)33 b Fp(t)25 b Fl(2)g Fp(S)550 5098 y Fn(\016)608 5083 y Fl([)20 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\))p Fm(,)34 b Fp(z)1073 5097 y Fo(1)1138 5083 y Fl(2)24 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\))p Fm(,)34 b Fp(z)1597 5097 y Fo(2)1662 5083 y Fl(2)25 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\))p Fm(,)75 5310 y Fq(\()p Fl(j)p Fp(v)s Fl(j)p Fq(\))1274 b Fl(j)p Fp(v)s Fq(\()p Fp(t;)15 b(z)1738 5324 y Fo(1)1779 5310 y Fp(;)g(z)1861 5324 y Fo(2)1901 5310 y Fq(\))p Fl(j)26 b(\024)2103 5287 y Fq(~)2083 5310 y Fp(C)6 b(e)2215 5256 y Fo(~)2196 5273 y Fn(K)f Fd(j)p Fn(t)p Fd(j)2326 5249 y Fg(m)2388 5310 y Fp(;)75 5521 y Fm(wher)-5 b(e)1445 5655 y Fp(m)25 b Fq(=)2030 5594 y(1)p 1656 5634 794 4 v 1656 5718 a(\()p Fp(l)1718 5732 y Fo(1)1758 5718 y Fp(=q)1844 5732 y Fo(1)1883 5718 y Fq(\))c(+)f(\()p Fp(l)2092 5732 y Fo(2)2132 5718 y Fp(=q)2218 5732 y Fo(2)2257 5718 y Fq(\))h Fl(\000)f Fq(1)2459 5655 y Fp(:)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 3764 100 a Fq(13)75 399 y Ff(Pro)s(of)75 564 y(Case)34 b Fp(l)352 578 y Fo(1)417 564 y Fl(6)p Fq(=)25 b(0)35 b Ff(and)g Fp(l)822 578 y Fo(2)887 564 y Fl(6)p Fq(=)25 b(0.)216 729 y(W)-8 b(e)34 b(consider)f(a)g(curv)m(e)g Fl(C)i Fq(=)29 b Fp(C)1310 696 y Fn(d)1303 785 y(\032)1339 747 y Fk(1)p Fg(=q)1429 762 y Fk(1)1339 806 y(1)1494 729 y Fl(\002)22 b Fp(C)1659 696 y Fn(d)1652 785 y(\032)1688 747 y Fk(1)p Fg(=q)1778 762 y Fk(2)1688 806 y(2)1854 729 y Fq(where)33 b Fp(C)2192 696 y Fn(d)2185 785 y(\032)2221 747 y Fk(1)p Fg(=q)2311 762 y Fk(1)2221 806 y(1)2387 729 y Fq(is)g(homotopic)h(to)f(the)h(circle)g Fp(C)3506 772 y Fn(\032)3542 735 y Fk(1)p Fg(=q)3632 750 y Fk(1)3542 794 y(1)3708 729 y Fq(and)75 894 y Fp(C)147 861 y Fn(d)140 951 y(\032)176 913 y Fk(1)p Fg(=q)266 928 y Fk(2)176 972 y(2)340 894 y Fq(is)c(homotopic)i(to)f(the)g(circle)g Fp(C)1446 938 y Fn(\032)1482 900 y Fk(1)p Fg(=q)1572 915 y Fk(2)1482 959 y(2)1646 894 y Fq(and)f(satis\014es)h(the)f(follo)m (wing)i(prop)s(erties.)75 1100 y Ff(i\))e Fq(F)-8 b(or)31 b(\()p Fp(v)417 1114 y Fo(1)457 1100 y Fp(;)15 b(v)541 1114 y Fo(2)581 1100 y Fq(\))25 b Fl(2)g(C)36 b Fq(suc)m(h)30 b(that)h(\()p Fp(v)1292 1114 y Fo(1)1332 1100 y Fp(;)15 b(v)1416 1114 y Fo(2)1456 1100 y Fq(\))25 b Fl(2)g(N)13 b Fq(,)31 b(w)m(e)f(ha)m(v)m(e)i(\()p Fp(v)2168 1114 y Fo(1)2208 1100 y Fp(;)15 b(v)2292 1114 y Fo(2)2332 1100 y Fq(\))25 b Fl(2)g Fp(C)2543 1144 y Fn(\032)2579 1106 y Fk(1)p Fg(=q)2669 1121 y Fk(1)2579 1165 y(1)2733 1100 y Fl(\002)20 b Fp(C)2889 1144 y Fn(\032)2925 1106 y Fk(1)p Fg(=q)3015 1121 y Fk(2)2925 1165 y(2)3059 1100 y Fq(,)75 1242 y Ff(ii\))32 b Fq(There)g(exists)i Fp(a)29 b(>)f Fq(0,)34 b Fp(b)29 b(>)g Fq(0,)34 b Fp(\017)29 b(>)f Fq(0,)34 b(suc)m(h)e(that)i(for)e(\()p Fp(v)2173 1256 y Fo(1)2213 1242 y Fp(;)15 b(v)2297 1256 y Fo(2)2337 1242 y Fq(\))29 b Fl(2)g(C)37 b Fq(with)c(\()p Fp(v)2865 1256 y Fo(1)2905 1242 y Fp(;)15 b(v)2989 1256 y Fo(2)3029 1242 y Fq(\))29 b Fl(2)f(N)3270 1209 y Fn(c)3338 1242 y Fq(\(where)k Fl(N)3726 1209 y Fn(c)3793 1242 y Fq(is)75 1355 y(the)f(complemen)m(t)g(of)g Fl(N)43 b Fq(in)30 b Fe(C)1133 1322 y Fo(2)1172 1355 y Fq(\),)h(w)m(e)g(ha)m(v)m(e)g(that) g(for)g(all)g(\()p Fp(z)2146 1369 y Fo(1)2186 1355 y Fp(;)15 b(z)2268 1369 y Fo(2)2308 1355 y Fq(\))26 b Fl(2)e Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\))22 b Fl(\002)e Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)514 1609 y(0)25 b Fp(<)1171 1548 y Fl(j)p Fp(t)p Fl(j)p 690 1588 1045 4 v 690 1675 a(j)p Fp(v)762 1631 y Fn(q)794 1640 y Fk(1)759 1701 y Fo(1)853 1675 y Fl(\000)20 b Fp(z)986 1689 y Fo(1)1026 1675 y Fl(j)1051 1649 y Fn(l)1072 1658 y Fk(1)1107 1649 y Fn(=q)1174 1658 y Fk(1)1212 1675 y Fl(j)p Fp(v)1284 1631 y Fn(q)1316 1640 y Fk(2)1281 1701 y Fo(2)1375 1675 y Fl(\000)g Fp(z)1508 1689 y Fo(2)1548 1675 y Fl(j)1573 1649 y Fn(l)1594 1658 y Fk(2)1629 1649 y Fn(=q)1696 1658 y Fk(2)1770 1609 y Fp(<)25 b(\032)60 b(;)76 b Fl(j)p Fp(v)2146 1565 y Fn(q)2178 1574 y Fk(1)2143 1635 y Fo(1)2217 1609 y Fl(j)26 b(\024)f Fp(a)p Fl(j)p Fp(t)p Fl(j)2495 1572 y Fn(q)2527 1581 y Fk(1)2561 1572 y Fn(=l)2617 1581 y Fk(1)2717 1609 y Fp(;)76 b Fl(j)p Fp(v)2890 1565 y Fn(q)2922 1574 y Fk(2)2887 1635 y Fo(2)2961 1609 y Fl(j)25 b(\024)g Fp(b)p Fl(j)p Fp(t)p Fl(j)3229 1572 y Fn(q)3261 1581 y Fk(2)3296 1572 y Fn(=l)3352 1581 y Fk(2)3391 1609 y Fp(:)75 1867 y Fq(F)-8 b(rom)31 b(the)f(latter)i(de\014nition,)e(w)m (e)h(notice)h(that)75 2032 y Fl(\001)f Fq(F)-8 b(or)31 b(\()p Fp(v)373 2046 y Fo(1)413 2032 y Fp(;)15 b(v)497 2046 y Fo(2)536 2032 y Fq(\))26 b Fl(2)f(C)35 b Fq(with)30 b(\()p Fp(v)1052 2046 y Fo(1)1092 2032 y Fp(;)15 b(v)1176 2046 y Fo(2)1216 2032 y Fq(\))26 b Fl(2)f(N)13 b Fq(,)30 b(w)m(e)h(get)1009 2274 y(arg)r(\()1708 2213 y Fp(t)p 1182 2253 1085 4 v 1182 2340 a Fq(\()p Fp(v)1264 2296 y Fn(q)1296 2305 y Fk(1)1261 2366 y Fo(1)1355 2340 y Fl(\000)20 b Fp(z)1488 2354 y Fo(1)1528 2340 y Fq(\))1563 2314 y Fn(l)1584 2323 y Fk(1)1619 2314 y Fn(=q)1686 2323 y Fk(1)1724 2340 y Fq(\()p Fp(v)1806 2296 y Fn(q)1838 2305 y Fk(2)1803 2366 y Fo(2)1898 2340 y Fl(\000)f Fp(z)2030 2354 y Fo(2)2070 2340 y Fq(\))2105 2314 y Fn(l)2126 2323 y Fk(2)2161 2314 y Fn(=q)2228 2323 y Fk(2)2277 2274 y Fq(\))25 b Fl(2)g Fq(\()p Fp(d)2505 2237 y Fo(1)2505 2297 y Fn(\025)2546 2306 y Fk(0)2582 2297 y Fn(;\016)2639 2274 y Fp(;)15 b(d)2726 2237 y Fo(2)2726 2297 y Fn(\025)2767 2306 y Fk(0)2802 2297 y Fn(;\016)2860 2274 y Fq(\))p Fp(;)75 2551 y Fq(for)36 b(all)g Fp(t)f Fl(2)f Fp(S)570 2566 y Fn(\016)607 2551 y Fq(.)57 b(So)36 b(that)h(if)f(w)m(e)g(put)f Fp(t)1457 2518 y Fd(0)1515 2551 y Fq(=)f Fp(t=)p Fq(\()p Fp(v)1780 2507 y Fn(q)1812 2516 y Fk(1)1777 2577 y Fo(1)1875 2551 y Fl(\000)23 b Fp(z)2011 2565 y Fo(1)2051 2551 y Fq(\))2086 2518 y Fn(l)2107 2527 y Fk(1)2142 2518 y Fn(=q)2209 2527 y Fk(1)2248 2551 y Fq(\()p Fp(v)2330 2507 y Fn(q)2362 2516 y Fk(2)2327 2577 y Fo(2)2424 2551 y Fl(\000)h Fp(z)2561 2565 y Fo(2)2601 2551 y Fq(\))2636 2518 y Fn(l)2657 2527 y Fk(2)2692 2518 y Fn(=q)2759 2527 y Fk(2)2797 2551 y Fq(,)38 b(w)m(e)e(ha)m(v)m(e)h Fp(t)3247 2518 y Fd(0)3305 2551 y Fl(2)d Fp(G)3471 2566 y Fn(\016)3509 2551 y Fq(\()p Fp(\025)3597 2565 y Fo(0)3637 2551 y Fq(\))i(and)75 2664 y Fp(k)s Fq(\()p Fp(t)193 2631 y Fd(0)217 2664 y Fp(;)15 b(u)309 2678 y Fo(1)349 2664 y Fp(;)g(u)441 2678 y Fo(2)480 2664 y Fq(\))31 b(is)f(w)m(ell)i(de\014ned)d(for)h(all)h(\()p Fp(u)1487 2678 y Fo(1)1527 2664 y Fp(;)15 b(u)1619 2678 y Fo(2)1659 2664 y Fq(\))26 b Fl(2)f Fe(C)1872 2631 y Fo(2)1911 2664 y Fq(.)75 2829 y Fl(\001)h Fq(If)e(w)m(e)i(put)f Fp(t)535 2796 y Fd(0)583 2829 y Fq(=)g Fp(t=)p Fq(\()p Fp(v)839 2785 y Fn(q)871 2794 y Fk(1)836 2855 y Fo(1)921 2829 y Fl(\000)10 b Fp(z)1044 2843 y Fo(1)1083 2829 y Fq(\))1118 2796 y Fn(l)1139 2805 y Fk(1)1174 2796 y Fn(=q)1241 2805 y Fk(1)1280 2829 y Fq(\()p Fp(v)1362 2785 y Fn(q)1394 2794 y Fk(2)1359 2855 y Fo(2)1443 2829 y Fl(\000)g Fp(z)1566 2843 y Fo(2)1606 2829 y Fq(\))1641 2796 y Fn(l)1662 2805 y Fk(2)1697 2796 y Fn(=q)1764 2805 y Fk(2)1802 2829 y Fq(,)27 b(for)e(\()p Fp(v)2067 2843 y Fo(1)2107 2829 y Fp(;)15 b(v)2191 2843 y Fo(2)2231 2829 y Fq(\))25 b Fl(2)g(C)30 b Fq(with)c(\()p Fp(v)2737 2843 y Fo(1)2776 2829 y Fp(;)15 b(v)2860 2843 y Fo(2)2900 2829 y Fq(\))26 b Fl(2)f(N)3135 2796 y Fn(c)3195 2829 y Fq(,)h(w)m(e)g(ha)m(v)m(e)h Fl(j)p Fp(t)3638 2796 y Fd(0)3661 2829 y Fl(j)f Fp(<)f(\032)75 2942 y Fq(and)30 b(the)g(function)g Fp(k)s Fq(\()p Fp(t)882 2909 y Fd(0)906 2942 y Fp(;)15 b(u)998 2956 y Fo(1)1038 2942 y Fp(;)g(u)1130 2956 y Fo(2)1170 2942 y Fq(\))30 b(is)h(w)m(ell)g(de\014ned)e(for)i(all)g(\()p Fp(u)2177 2956 y Fo(1)2217 2942 y Fp(;)15 b(u)2309 2956 y Fo(2)2348 2942 y Fq(\))26 b Fl(2)f Fe(C)2561 2909 y Fo(2)2600 2942 y Fq(.)216 3107 y(On)37 b(the)h(other)g(hand,)g(the)g(v)-5 b(alue)38 b(of)g(the)g(in)m(tegral)h(represen)m(tation)g(in)e(the)h (prop)s(osition)f(6)h(do)s(es)f(not)75 3220 y(dep)s(end)20 b(on)h(the)h(deformation)g(of)g(the)f(curv)m(e)h Fp(C)1687 3264 y Fn(\032)1723 3226 y Fk(1)p Fg(=q)1813 3241 y Fk(1)1723 3285 y(1)1859 3220 y Fl(\002)s Fp(C)1998 3264 y Fn(\032)2034 3226 y Fk(1)p Fg(=q)2124 3241 y Fk(2)2034 3285 y(2)2189 3220 y Fq(to)g(the)g(curv)m(e)g Fl(C)5 b Fq(,)23 b(b)s(ecause)f(they)f (are)h(homotopic)75 3361 y(to)38 b(eac)m(h)g(other.)61 b(Therefore,)38 b(w)m(e)g(deduce)e(an)h(analytic)i(con)m(tin)m(uation)g (of)e(the)g(function)g Fp(v)s Fq(\()p Fp(t;)15 b(z)3443 3375 y Fo(1)3483 3361 y Fp(;)g(z)3565 3375 y Fo(2)3605 3361 y Fq(\))38 b(with)75 3474 y(resp)s(ect)30 b(to)h Fp(t)26 b Fl(2)e Fp(S)697 3489 y Fn(\016)755 3474 y Fl([)c Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\),)32 b Fp(z)1215 3488 y Fo(1)1280 3474 y Fl(2)25 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)32 b Fp(z)1735 3488 y Fo(2)1800 3474 y Fl(2)24 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\))32 b(whic)m(h)e(will)h(b)s(e)f(also)h(denoted)f(b)m(y)582 3747 y Fp(v)s Fq(\()p Fp(t;)15 b(z)779 3761 y Fo(1)819 3747 y Fp(;)g(z)901 3761 y Fo(2)942 3747 y Fq(\))25 b(=)1206 3686 y(1)p 1108 3727 242 4 v 1108 3810 a(\(2)p Fp(\031)s(i)p Fq(\))1309 3784 y Fo(2)1360 3747 y Fp(@)1408 3761 y Fn(z)1441 3770 y Fk(1)1480 3747 y Fp(@)1528 3761 y Fn(z)1561 3770 y Fk(2)1615 3624 y Fh(I)1665 3830 y Fd(C)1726 3747 y Fq(\010)19 b Fl(\003)i Fp(k)s Fq(\()p Fp(t;)15 b(v)2079 3761 y Fo(1)2119 3747 y Fp(;)g(v)2203 3761 y Fo(2)2243 3747 y Fq(\))2462 3686 y Fp(v)2509 3642 y Fn(q)2541 3651 y Fk(1)2575 3642 y Fd(\000)p Fo(1)2506 3712 y(1)2670 3686 y Fp(v)2717 3642 y Fn(q)2749 3651 y Fk(2)2783 3642 y Fd(\000)p Fo(1)2714 3712 y(2)p 2288 3727 763 4 v 2288 3810 a Fq(\()p Fp(v)2370 3766 y Fn(q)2402 3775 y Fk(1)2367 3836 y Fo(1)2462 3810 y Fl(\000)20 b Fp(z)2595 3824 y Fo(1)2634 3810 y Fq(\)\()p Fp(v)2751 3766 y Fn(q)2783 3775 y Fk(2)2748 3836 y Fo(2)2843 3810 y Fl(\000)g Fp(z)2976 3824 y Fo(2)3015 3810 y Fq(\))3061 3747 y Fp(dv)3152 3761 y Fo(1)3192 3747 y Fp(dv)3283 3761 y Fo(2)3322 3747 y Fp(:)75 4006 y Fq(The)k(curv)m(e)g Fl(C)29 b Fq(can)c(b)s(e)e (written)i(as)f(the)g(union)g(of)g(t)m(w)m(o)h(disjoin)m(ted)g(parts)f Fl(C)2608 4020 y Fo(1)2672 4006 y Fq(=)h Fl(f)p Fq(\()p Fp(v)2892 4020 y Fo(1)2933 4006 y Fp(;)15 b(v)3017 4020 y Fo(2)3056 4006 y Fq(\))26 b Fl(2)f(C)30 b Fq(:)c(\()p Fp(v)3411 4020 y Fo(1)3451 4006 y Fp(;)15 b(v)3535 4020 y Fo(2)3575 4006 y Fq(\))25 b Fl(2)g(N)13 b(g)75 4119 y Fq(,)31 b Fl(C)179 4133 y Fo(2)243 4119 y Fq(=)25 b Fl(f)p Fq(\()p Fp(v)463 4133 y Fo(1)503 4119 y Fp(;)15 b(v)587 4133 y Fo(2)627 4119 y Fq(\))26 b Fl(2)f(C)30 b Fq(:)c(\()p Fp(v)982 4133 y Fo(1)1022 4119 y Fp(;)15 b(v)1106 4133 y Fo(2)1145 4119 y Fq(\))26 b Fl(2)f(N)1380 4086 y Fn(c)1415 4119 y Fl(g)p Fq(.)41 b(In)29 b(the)i(follo)m(wing,)h (w)m(e)f(put)536 4392 y Fp(A)604 4406 y Fo(1)644 4392 y Fq(\()p Fp(t;)15 b(z)794 4406 y Fo(1)834 4392 y Fp(;)g(z)916 4406 y Fo(2)956 4392 y Fq(\))26 b(=)1221 4331 y(1)p 1123 4372 242 4 v 1123 4455 a(\(2)p Fp(\031)s(i)p Fq(\))1324 4429 y Fo(2)1375 4392 y Fp(@)1423 4406 y Fn(z)1456 4415 y Fk(1)1495 4392 y Fp(@)1543 4406 y Fn(z)1576 4415 y Fk(2)1630 4269 y Fh(I)1680 4475 y Fd(C)1717 4484 y Fk(1)1771 4392 y Fq(\010)20 b Fl(\003)h Fp(k)s Fq(\()p Fp(t;)15 b(v)2125 4406 y Fo(1)2165 4392 y Fp(;)g(v)2249 4406 y Fo(2)2289 4392 y Fq(\))2508 4331 y Fp(v)2555 4287 y Fn(q)2587 4296 y Fk(1)2621 4287 y Fd(\000)p Fo(1)2552 4357 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Fq(\))2508 4747 y Fp(v)2555 4703 y Fn(q)2587 4712 y Fk(1)2621 4703 y Fd(\000)p Fo(1)2552 4773 y(1)2715 4747 y Fp(v)2762 4703 y Fn(q)2794 4712 y Fk(2)2829 4703 y Fd(\000)p Fo(1)2759 4773 y(2)p 2334 4787 763 4 v 2334 4871 a Fq(\()p Fp(v)2416 4827 y Fn(q)2448 4836 y Fk(1)2413 4897 y Fo(1)2507 4871 y Fl(\000)20 b Fp(z)2640 4885 y Fo(1)2680 4871 y Fq(\)\()p Fp(v)2797 4827 y Fn(q)2829 4836 y Fk(2)2794 4897 y Fo(2)2889 4871 y Fl(\000)f Fp(z)3021 4885 y Fo(2)3061 4871 y Fq(\))3106 4808 y Fp(dv)3197 4822 y Fo(1)3237 4808 y Fp(dv)3328 4822 y Fo(2)3368 4808 y Fp(;)75 5033 y Fq(so,)31 b(that)g Fp(v)s Fq(\()p Fp(t;)15 b(z)606 5047 y Fo(1)646 5033 y Fp(;)g(z)728 5047 y Fo(2)769 5033 y Fq(\))25 b(=)g Fp(A)993 5047 y Fo(1)1033 5033 y Fq(\()p Fp(t;)15 b(z)1183 5047 y Fo(1)1223 5033 y Fp(;)g(z)1305 5047 y Fo(2)1345 5033 y Fq(\))21 b(+)f Fp(A)1560 5047 y Fo(2)1599 5033 y Fq(\()p Fp(t;)15 b(z)1749 5047 y Fo(1)1790 5033 y Fp(;)g(z)1872 5047 y Fo(2)1912 5033 y Fq(\).)216 5198 y(Due)32 b(to)f(the)h(con)m(tin)m(uit)m(y)h(of)e(the)h(function)e (\010)h(and)g(the)g(estimates)i(of)e(the)h(prop)s(osition)e(5,)i(there) g(exists)75 5311 y Fp(K)159 5278 y Fo(1)223 5311 y Fp(>)25 b Fq(0)d(with)f(the)h(follo)m(wing)h(prop)s(erties.)37 b(F)-8 b(or)22 b Fp(t)j Fl(2)g Fp(S)1921 5326 y Fn(\016)1961 5311 y Fl([)r Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\),)24 b Fp(z)2395 5325 y Fo(1)2460 5311 y Fl(2)h Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)25 b Fp(z)2908 5325 y Fo(2)2973 5311 y Fl(2)f Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)25 b(\()p Fp(v)3457 5325 y Fo(1)3497 5311 y Fp(;)15 b(v)3581 5325 y Fo(2)3621 5311 y Fq(\))25 b Fl(2)g(C)3815 5325 y Fo(1)75 5424 y Fq(\()31 b Fl(j)p Fp(v)210 5438 y Fo(1)249 5424 y Fl(j)274 5391 y Fn(q)306 5400 y Fk(1)370 5424 y Fq(=)25 b Fp(\032)513 5438 y Fo(1)553 5424 y Fq(,)30 b Fl(j)p Fp(v)677 5438 y Fo(2)717 5424 y Fl(j)742 5391 y Fn(q)774 5400 y Fk(2)838 5424 y Fq(=)25 b Fp(\032)981 5438 y Fo(2)1020 5424 y Fq(,)31 b(0)25 b Fl(\024)g(j)p Fp(u)1319 5438 y 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5660 y Fo(1)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 75 100 a Fq(14)75 399 y(and)738 535 y Fl(j)p Fp(k)s Fq(\()1384 473 y Fp(t)p 858 514 1085 4 v 858 601 a Fq(\()p Fp(v)940 557 y Fn(q)972 566 y Fk(1)937 627 y Fo(1)1032 601 y Fl(\000)19 b Fp(z)1164 615 y Fo(1)1204 601 y Fq(\))1239 575 y Fn(l)1260 584 y Fk(1)1295 575 y Fn(=q)1362 584 y Fk(1)1401 601 y Fq(\()p Fp(v)1483 557 y Fn(q)1515 566 y Fk(2)1480 627 y Fo(2)1574 601 y Fl(\000)h Fp(z)1707 615 y Fo(2)1746 601 y Fq(\))1781 575 y Fn(l)1802 584 y Fk(2)1837 575 y Fn(=q)1904 584 y Fk(2)1953 535 y Fp(;)15 b(u)2045 549 y Fo(1)2085 535 y Fp(;)g(u)2177 549 y Fo(2)2217 535 y Fq(\))p Fl(j)26 b(\024)f Fp(C)2464 549 y Fo(3)2503 535 y Fp(e)2545 497 y Fn(R)2598 506 y Fk(1)2633 497 y Fn(u)2674 506 y Fk(1)2713 535 y Fp(e)2755 497 y Fn(R)2808 506 y Fk(2)2843 497 y Fn(u)2884 506 y Fk(2)2948 535 y Fl(\024)f Fp(K)3127 497 y Fo(1)3167 535 y Fp(:)75 785 y Fq(F)-8 b(rom)32 b(the)f(latter)h (estimates,)h(w)m(e)f(deduce)f(that)g(there)h(exists)2292 762 y(~)2267 785 y Fp(K)2344 799 y Fo(1)2410 785 y Fp(>)27 b Fq(0,)2630 762 y(~)2610 785 y Fp(C)32 b(>)27 b Fq(0)k(and)3083 762 y(~)3059 785 y Fp(K)i(>)26 b Fq(0)31 b(suc)m(h)g(that)h(for)75 898 y(all)f Fp(t)25 b Fl(2)g Fp(S)401 913 y Fn(\016)459 898 y Fl([)20 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\),)32 b Fp(z)919 912 y Fo(1)984 898 y Fl(2)25 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)31 b Fp(z)1438 912 y Fo(2)1503 898 y Fl(2)25 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)75 1130 y(\()p Fp(A)178 1144 y Fo(1)218 1130 y Fq(\))1114 b Fl(j)p Fp(A)1460 1144 y Fo(1)1500 1130 y Fq(\()p Fp(t;)15 b(z)1650 1144 y Fo(1)1690 1130 y Fp(;)g(z)1772 1144 y Fo(2)1812 1130 y Fq(\))p Fl(j)26 b(\024)2018 1107 y Fq(~)1994 1130 y Fp(K)2071 1144 y Fo(1)2136 1130 y Fl(\024)2252 1107 y Fq(~)2232 1130 y Fp(C)6 b(e)2364 1076 y Fo(~)2345 1093 y Fn(K)f Fd(j)p Fn(t)p Fd(j)2475 1069 y Fg(m)2537 1130 y Fp(:)216 1348 y Fq(F)-8 b(rom)41 b(the)g(assumption)e(\()p Fp(C)7 b(d)p Fq(3\))42 b(and)d(the)i(estimates)h(in)e(the)g(prop)s (osition)g(5,)k(there)d(exist)g Fp(K)3597 1315 y Fd(0)3590 1373 y Fo(1)3671 1348 y Fp(>)h Fq(0,)75 1461 y Fp(K)159 1428 y Fd(0)152 1486 y Fo(2)232 1461 y Fp(>)e Fq(0,)479 1438 y(~)455 1461 y Fp(K)539 1428 y Fd(0)532 1486 y Fo(1)612 1461 y Fp(>)g Fq(0,)i Fp(M)51 b(>)40 b Fq(0)g(with)f(the)g(follo)m (wing)i(prop)s(erties.)67 b(F)-8 b(or)40 b Fp(t)g Fl(2)g Fp(S)2836 1476 y Fn(\016)2900 1461 y Fl([)26 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\),)43 b Fp(z)3377 1475 y Fo(1)3457 1461 y Fl(2)d Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)75 1574 y Fp(z)117 1588 y Fo(2)183 1574 y Fl(2)25 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)33 b(\()p Fp(v)676 1588 y Fo(1)715 1574 y Fp(;)15 b(v)799 1588 y Fo(2)839 1574 y Fq(\))27 b Fl(2)e(C)1035 1588 y Fo(2)1105 1574 y Fq(\()31 b Fl(j)p Fp(v)1240 1588 y Fo(1)1280 1574 y Fl(j)1305 1541 y Fn(q)1337 1550 y Fk(1)1401 1574 y Fl(\024)26 b Fp(a)p Fl(j)p Fp(t)p Fl(j)1629 1541 y Fn(q)1661 1550 y Fk(1)1696 1541 y Fn(=l)1752 1550 y Fk(1)1791 1574 y Fq(,)31 b Fl(j)p Fp(v)1916 1588 y Fo(2)1956 1574 y Fl(j)1981 1541 y Fn(q)2013 1550 y Fk(2)2077 1574 y Fl(\024)26 b Fp(b)p Fl(j)p Fp(t)p Fl(j)2296 1541 y Fn(q)2328 1550 y Fk(2)2362 1541 y Fn(=l)2418 1550 y Fk(2)2457 1574 y Fq(,)31 b Fp(u)2565 1588 y Fo(1)2631 1574 y Fq(=)25 b Fp(v)2774 1530 y Fn(q)2806 1539 y Fk(1)2771 1600 y Fo(1)2845 1574 y Fp(s)2888 1588 y Fo(1)2927 1574 y Fq(,)31 b Fp(s)3026 1588 y Fo(1)3091 1574 y Fl(2)26 b Fq([0)p Fp(;)15 b Fq(1],)33 b Fp(u)3468 1588 y Fo(2)3533 1574 y Fq(=)26 b Fp(v)3677 1530 y Fn(q)3709 1539 y Fk(2)3674 1600 y Fo(1)3747 1574 y Fp(s)3790 1588 y Fo(2)3829 1574 y Fq(,)75 1687 y Fp(s)118 1701 y Fo(2)182 1687 y Fl(2)f Fq([0)p Fp(;)15 b Fq(1])32 b(\),)f(w)m(e)g(put)654 1937 y Fp(A)25 b Fq(=)g Fl(j)p Fq(\010\()p Fp(v)1016 1893 y Fn(q)1048 1902 y Fk(1)1013 1963 y Fo(1)1107 1937 y Fl(\000)20 b Fp(u)1250 1951 y Fo(1)1290 1937 y Fp(;)15 b(v)1377 1893 y Fn(q)1409 1902 y Fk(2)1374 1963 y Fo(2)1468 1937 y Fl(\000)20 b Fp(u)1611 1951 y Fo(2)1650 1937 y Fq(\))p Fl(jj)p Fp(k)s Fq(\()2357 1876 y Fp(t)p 1830 1917 V 1830 2004 a Fq(\()p Fp(v)1912 1959 y Fn(q)1944 1968 y Fk(1)1909 2030 y Fo(1)2005 2004 y Fl(\000)g Fp(z)2138 2018 y Fo(1)2177 2004 y Fq(\))2212 1977 y Fn(l)2233 1986 y Fk(1)2268 1977 y Fn(=q)2335 1986 y Fk(1)2374 2004 y Fq(\()p Fp(v)2456 1959 y Fn(q)2488 1968 y Fk(2)2453 2030 y Fo(2)2547 2004 y Fl(\000)g Fp(z)2680 2018 y Fo(2)2720 2004 y Fq(\))2755 1977 y Fn(l)2776 1986 y Fk(2)2811 1977 y Fn(=q)2878 1986 y Fk(2)2926 1937 y Fp(;)15 b(u)3018 1951 y Fo(1)3058 1937 y Fp(;)g(u)3150 1951 y Fo(2)3190 1937 y Fq(\))p Fl(j)p Fp(:)75 2202 y Fq(W)-8 b(e)32 b(get)620 2326 y Fp(A)26 b Fl(\024)f Fp(C)7 b(C)947 2340 y Fo(3)1001 2326 y Fq(exp\()p Fp(K)1252 2340 y Fo(1)1292 2326 y Fl(j)p Fp(v)1364 2281 y Fn(q)1396 2290 y Fk(1)1361 2352 y Fo(1)1455 2326 y Fl(\000)20 b Fp(u)1598 2340 y Fo(1)1637 2326 y Fl(j)1662 2288 y Fn(k)1699 2297 y Fk(1)1758 2326 y Fq(+)g Fp(R)1918 2340 y Fo(1)1957 2326 y Fl(j)p Fp(u)2034 2340 y Fo(1)2074 2326 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a(\(2)p Fp(\031)s(i)p Fq(\))1324 2015 y Fo(2)1375 1979 y Fp(@)1423 1993 y Fn(z)1456 2002 y Fk(1)1495 1979 y Fp(@)1543 1993 y Fn(z)1576 2002 y Fk(2)1630 1855 y Fh(I)1680 2061 y Fd(C)1717 2070 y Fk(2)1771 1979 y Fq(\010)20 b Fl(\003)h Fp(k)s Fq(\()p Fp(t;)15 b(v)2125 1993 y Fo(1)2165 1979 y Fp(;)g(v)2249 1993 y Fo(2)2289 1979 y Fq(\))2508 1917 y Fp(v)2555 1873 y Fn(q)2587 1882 y Fk(1)2621 1873 y Fd(\000)p Fo(1)2552 1943 y(1)2715 1917 y Fp(v)2762 1873 y Fn(q)2794 1882 y Fk(2)2829 1873 y Fd(\000)p Fo(1)2759 1943 y(2)p 2334 1958 763 4 v 2334 2042 a Fq(\()p Fp(v)2416 1997 y Fn(q)2448 2006 y Fk(1)2413 2068 y Fo(1)2507 2042 y Fl(\000)20 b Fp(z)2640 2056 y Fo(1)2680 2042 y Fq(\)\()p Fp(v)2797 1997 y Fn(q)2829 2006 y Fk(2)2794 2068 y Fo(2)2889 2042 y Fl(\000)f Fp(z)3021 2056 y Fo(2)3061 2042 y Fq(\))3106 1979 y Fp(dv)3197 1993 y Fo(1)3237 1979 y Fp(dv)3328 1993 y Fo(2)3368 1979 y Fp(;)75 2205 y Fq(so,)31 b(that)g Fp(v)s Fq(\()p Fp(t;)15 b(z)606 2219 y Fo(1)646 2205 y Fp(;)g(z)728 2219 y Fo(2)769 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Fk(2)1645 3416 y Fo(2)1739 3390 y Fl(\000)20 b Fp(u)1882 3404 y Fo(2)1921 3390 y Fq(\))p Fl(jj)p Fp(k)s Fq(\()2357 3329 y Fp(t)p 2101 3369 543 4 v 2101 3456 a Fq(\()p Fp(v)2183 3412 y Fn(q)2215 3421 y Fk(1)2180 3482 y Fo(1)2276 3456 y Fl(\000)g Fp(z)2409 3470 y Fo(1)2448 3456 y Fq(\))2483 3430 y Fn(l)2504 3439 y Fk(1)2539 3430 y Fn(=q)2606 3439 y Fk(1)2655 3390 y Fp(;)15 b(u)2747 3404 y Fo(1)2787 3390 y Fp(;)g(u)2879 3404 y Fo(2)2919 3390 y Fq(\))p Fl(j)p Fp(:)75 3654 y Fq(F)-8 b(rom)31 b(the)f(assumption)g(\()p Fp(C)7 b(d)p Fq(3\))32 b(and)d(the)i(estimates)h(in)e(the)g(prop)s (osition)h(5,)g(w)m(e)f(get)831 3862 y Fp(A)c Fl(\024)f Fp(C)7 b(C)1158 3876 y Fo(3)1197 3862 y Fp(K)1281 3824 y Fd(0)1274 3884 y Fo(2)1328 3862 y Fq(exp\()p Fp(K)1579 3876 y Fo(1)1619 3862 y Fp(a)1667 3824 y Fn(k)1704 3833 y Fk(1)1743 3862 y Fl(j)p Fp(t)p Fl(j)1826 3824 y Fn(k)1863 3833 y Fk(1)1897 3824 y Fn(q)1929 3833 y Fk(1)1964 3824 y Fn(=l)2020 3833 y Fk(1)2059 3862 y Fq(\(1)21 b Fl(\000)f Fp(s)2294 3876 y 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Fp(K)38 b(>)32 b Fq(0)j(suc)m(h)f(that)h(for)f (all)h Fp(t)d Fl(2)f Fp(S)3404 4501 y Fn(\016)3464 4486 y Fl([)23 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\),)75 4599 y Fp(z)117 4613 y Fo(1)182 4599 y Fl(2)25 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)32 b Fp(z)637 4613 y Fo(2)702 4599 y Fl(2)24 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)p Fq(\),)75 4807 y(\()p Fp(A)178 4774 y Fd(0)178 4831 y Fo(2)218 4807 y Fq(\))1121 b Fl(j)p Fp(A)1467 4821 y Fo(2)1507 4807 y Fq(\()p Fp(t;)15 b(z)1657 4821 y Fo(1)1697 4807 y Fp(;)g(z)1779 4821 y Fo(2)1819 4807 y Fq(\))p Fl(j)26 b(\024)2021 4784 y Fq(~)2001 4807 y Fp(C)21 b Fq(exp\()2286 4784 y(~)2261 4807 y Fp(K)7 b Fl(j)p Fp(t)p Fl(j)2428 4769 y Fn(m)2495 4807 y Fq(\))p Fp(:)75 5015 y Fq(In)30 b(conclusion,)h(from)f(the)g(estimates)i(\()p Fp(A)1528 4982 y Fd(0)1528 5039 y Fo(1)1568 5015 y Fq(\))f(and)f(\()p Fp(A)1914 4982 y Fd(0)1914 5039 y Fo(2)1954 5015 y Fq(\),)h(w)m(e)f (deduce)g(the)h(estimate)h(\()p Fl(j)p Fp(v)s Fl(j)p Fq(\).)562 b Fc(2)75 5317 y Fb(3.3)112 b(Summabilit)m(y)40 b(of)d(formal)i(solutions)75 5491 y Ff(De\014nition)c(1)46 b Fm(A)32 b(series)1411 5655 y Fq(^)-51 b Fp(u)p Fq(\()p Fp(t;)15 b(z)1607 5669 y Fo(1)1647 5655 y Fp(;)g(z)1729 5669 y Fo(2)1769 5655 y Fq(\))26 b(=)1956 5541 y Fd(1)1926 5568 y Fh(X)1930 5764 y Fn(j)t Fo(=0)2082 5593 y Fp(t)2115 5560 y Fn(j)p 2082 5634 70 4 v 2083 5717 a Fp(j)5 b Fq(!)2162 5655 y Fp(u)2214 5669 y Fn(j)2250 5655 y Fq(\()p Fp(z)2327 5669 y Fo(1)2367 5655 y Fp(;)15 b(z)2449 5669 y Fo(2)2489 5655 y Fq(\))p eop end %%Page: 16 16 TeXDict begin 16 15 bop 75 100 a Fq(16)75 399 y Fm(wher)-5 b(e)38 b Fp(u)388 413 y Fn(j)425 399 y Fq(\()p Fp(z)502 413 y Fo(1)542 399 y Fp(;)15 b(z)624 413 y Fo(2)664 399 y Fq(\))p Fm(,)38 b Fq(0)c Fl(\024)f Fp(j)39 b Fl(\024)33 b(1)k Fm(ar)-5 b(e)38 b(holomorphic)i(functions)e(in)f(a)g(p)-5 b(olydisc)39 b Fp(D)2956 413 y Fn(\032)2992 422 y Fk(1)3055 399 y Fl(\002)23 b Fp(D)3224 413 y Fn(\032)3260 422 y Fk(2)3299 399 y Fm(,)38 b(is)f(said)h(to)g(b)-5 b(e)75 511 y Fp(k)s 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b(onver)g(gent)34 b(for)f Fl(j)p Fp(z)1250 1241 y Fo(1)1290 1227 y Fl(j)26 b Fp(<)f(r)1478 1241 y Fo(1)1517 1227 y Fm(,)32 b Fl(j)p Fp(z)1644 1241 y Fo(2)1684 1227 y Fl(j)26 b Fp(<)f(r)1872 1241 y Fo(2)1944 1227 y Fm(and)33 b Fl(j)p Fp(t)p Fl(j)26 b Fp(<)f(R)q Fq(\()p Fp(r)2471 1241 y Fo(1)2511 1227 y Fp(;)15 b(r)2592 1241 y Fo(2)2631 1227 y Fq(\))p Fm(,)216 1390 y Ff(ii\))36 b Fm(Ther)-5 b(e)37 b(exists)g Fp(\016)f(>)c Fq(0)37 b Fm(such)f(that)i(for)f(al)5 b(l)37 b Fq(\()p Fp(z)1890 1404 y Fo(1)1930 1390 y Fp(;)15 b(z)2012 1404 y Fo(2)2052 1390 y Fq(\))33 b Fl(2)2234 1367 y Fq(\026)2212 1390 y Fp(D)s Fq(\(0)p Fp(;)15 b(r)2451 1404 y Fo(1)2492 1390 y Fq(\))23 b Fl(\002)2665 1367 y Fq(\026)2644 1390 y Fp(D)s Fq(\(0)p Fp(;)15 b(r)2883 1404 y Fo(2)2923 1390 y Fq(\))p Fm(,)38 b(the)f(series)f Fp(w)r Fq(\()p Fp(t;)15 b(z)3656 1404 y Fo(1)3697 1390 y Fp(;)g(z)3779 1404 y Fo(2)3819 1390 y Fq(\))75 1503 y Fm(c)-5 b(an)41 b(b)-5 b(e)40 b(analytic)-5 b(al)5 b(ly)43 b(c)-5 b(ontinue)g(d)42 b(with)f(r)-5 b(esp)g(e)g(ct)42 b(to)f Fp(t)f Fm(in)g(a)h(se)-5 b(ctor)41 b Fp(S)2507 1518 y Fn(d;\016)2640 1503 y Fq(=)e Fl(f)p Fp(t)g Fl(2)g Fe(C)g Fq(:)h Fl(j)p Fp(d)26 b Fl(\000)g Fq(arg)q(\()p Fp(t)p Fq(\))p Fl(j)40 b Fp(<)f(\016)s Fl(g)p Fm(.)75 1616 y(Mor)-5 b(e)g(over,)34 b(ther)-5 b(e)33 b(exist)g Fp(\016)981 1630 y Fo(1)1046 1616 y Fp(<)25 b(\016)s Fm(,)33 b Fp(C)f(>)25 b Fq(0)p Fm(,)33 b(and)g Fp(K)f(>)25 b Fq(0)33 b Fm(such)g(that)1453 1829 y Fq(sup)1283 1912 y Fd(j)p Fn(z)1336 1921 y Fk(1)1370 1912 y Fd(j\024)p Fn(r)1477 1921 y Fk(1)1511 1912 y Fn(;)p Fd(j)p Fn(z)1584 1921 y Fk(2)1618 1912 y Fd(j\024)p Fn(r)1725 1921 y Fk(2)1774 1829 y Fl(j)p Fp(w)r Fq(\()p Fp(t;)15 b(z)2016 1843 y Fo(1)2057 1829 y Fp(;)g(z)2139 1843 y Fo(2)2179 1829 y Fq(\))p Fl(j)26 b(\024)f Fp(C)7 b(e)2475 1792 y Fn(K)e Fd(j)p Fn(t)p Fd(j)2605 1768 y Fg(k)75 2100 y Fm(for)33 b(al)5 b(l)33 b Fp(t)25 b Fl(2)g Fp(S)550 2115 y Fn(d;\016)637 2124 y Fk(1)676 2100 y Fm(.)75 2314 y Fq(If)30 b(this)g(is)h(so,)f(the)h(Laplace)h(transform)d(of)i(order)f Fp(k)j Fq(of)e Fp(w)r Fq(\()p Fp(t;)15 b(z)2223 2328 y Fo(1)2264 2314 y Fp(;)g(z)2346 2328 y Fo(2)2386 2314 y Fq(\),)1056 2584 y Fp(u)p Fq(\()p Fp(t;)g(z)1258 2598 y Fo(1)1299 2584 y Fp(;)g(z)1381 2598 y Fo(2)1421 2584 y Fq(\))25 b(=)g Fp(t)1610 2546 y Fd(\000)p Fn(k)1723 2460 y Fh(Z)1814 2486 y Fd(1)p Fo(\()p Fn(\015)t Fo(\))1773 2666 y(0)1999 2584 y Fp(w)r Fq(\()p Fp(t;)15 b(z)2216 2598 y Fo(1)2256 2584 y Fp(;)g(z)2338 2598 y Fo(2)2379 2584 y Fq(\))p Fp(e)2456 2546 y Fd(\000)p Fo(\()p Fn(\034)8 b(=t)p Fo(\))2664 2523 y Fg(k)2708 2584 y Fp(d\034)2805 2546 y Fn(k)2848 2584 y Fp(;)75 2834 y Fq(where)32 b(the)h(in)m (tegration)h(is)f(made)f(along)i(the)e(ra)m(y)h(arg)r(\()p Fp(\034)10 b Fq(\))29 b(=)g Fp(\015)37 b Fq(with)32 b Fl(j)p Fp(d)23 b Fl(\000)e Fp(\015)5 b Fl(j)29 b Fp(<)f(\016)s Fq(,)34 b(is)f(called)g(the)g Fp(k)s Fl(\000)f Fq(sum)75 2947 y(of)f(the)f(formal)h(series)36 b(^)-51 b Fp(u)p Fq(\()p Fp(t;)15 b(z)1066 2961 y Fo(1)1107 2947 y Fp(;)g(z)1189 2961 y Fo(2)1229 2947 y Fq(\).)216 3110 y(W)-8 b(e)32 b(consider)e(the)h(formal)f(Borel)i(transform)1341 3361 y Fp(w)r Fq(\()p Fp(t;)15 b(z)1558 3375 y Fo(1)1599 3361 y Fp(;)g(z)1681 3375 y Fo(2)1721 3361 y Fq(\))25 b(=)1908 3248 y Fd(1)1877 3275 y Fh(X)1882 3470 y Fn(j)t Fo(=0)2077 3299 y Fp(u)2129 3313 y Fn(j)2166 3299 y Fq(\()p Fp(z)2243 3313 y Fo(1)2283 3299 y Fp(;)15 b(z)2365 3313 y Fo(2)2405 3299 y Fq(\))p Fp(t)2473 3266 y Fn(j)p 2034 3340 520 4 v 2034 3424 a Fp(j)5 b Fq(!\000\(1)21 b(+)f Fp(j)5 b(=m)p Fq(\))2564 3361 y Fp(;)75 3668 y Fq(where)26 b Fp(u)386 3682 y Fn(j)422 3668 y Fq(\()p Fp(z)499 3682 y Fo(1)539 3668 y Fp(;)15 b(z)621 3682 y Fo(2)661 3668 y Fq(\))27 b(is)f(de\014ned)f(b)m(y)h(the)g(form)m(ula)g(\()p Fp(u)1809 3682 y Fn(j)1846 3668 y Fq(\))g(in)g(the)g(section)i(1)e(and) g Fp(m)f Fq(=)g(\(\()p Fp(l)3003 3682 y Fo(1)3043 3668 y Fp(=q)3129 3682 y Fo(1)3168 3668 y Fq(\))12 b(+)g(\()p Fp(l)3360 3682 y Fo(2)3399 3668 y Fp(=q)3485 3682 y Fo(2)3525 3668 y Fq(\))g Fl(\000)g Fq(1\))3735 3635 y Fd(\000)p Fo(1)3829 3668 y Fq(.)75 3781 y(Due)37 b(to)g(the)g(general)h(theory)f (of)g(momen)m(t)g(summabilit)m(y)g(\(see)g([3],)j(p.)59 b(107{109\),)42 b(w)m(e)37 b(deduce)f(from)g(the)75 3894 y(prop)s(osition)30 b(7)h(that)g(there)f(exists)h(a)g(direction)g Fp(d)25 b Fl(2)g Fe(R)30 b Fq(and)g Fp(\016)f(>)c Fq(0)31 b(with)f Fp(S)2699 3909 y Fn(d;\016)2817 3894 y Fl(\032)25 b Fp(S)2969 3909 y Fn(\016)3037 3894 y Fq(suc)m(h)30 b(that)h Fp(w)r Fq(\()p Fp(t;)15 b(z)3656 3908 y Fo(1)3697 3894 y Fp(;)g(z)3779 3908 y Fo(2)3819 3894 y Fq(\))75 4007 y(satis\014es)32 b(the)f(prop)s(erties)g Ff(i)p Fq(\))h(and)e Ff(ii)p Fq(\))i(of)f(the)h(latter)h(de\014nition.)43 b(Finally)-8 b(,)33 b(w)m(e)f(get)g(the)g(main)f(result)g(of)h(this)75 4120 y(section.)75 4333 y Ff(Theorem)j(2)46 b Fm(L)-5 b(et)33 b Fp(m)27 b Fq(=)g(\(\()p Fp(l)1076 4347 y Fo(1)1116 4333 y Fp(=q)1202 4347 y Fo(1)1241 4333 y Fq(\))21 b(+)g(\()p Fp(l)1451 4347 y Fo(2)1491 4333 y Fp(=q)1577 4347 y Fo(2)1616 4333 y Fq(\))h Fl(\000)e Fq(1\))1844 4300 y Fd(\000)p Fo(1)1939 4333 y Fm(.)45 b(Under)33 b(the)h(c)-5 b(onditions)35 b Fq(\()p Fp(C)7 b(d)p Fq(1\))p Fm(,)35 b Fq(\()p Fp(C)7 b(d)p Fq(2\))34 b Fm(and)h Fq(\()p Fp(C)7 b(d)p Fq(3\))75 4446 y Fm(ther)-5 b(e)34 b(exists)e(a)h(dir)-5 b(e)g(ction)35 b Fp(d)25 b Fl(2)g Fe(R)32 b Fm(and)i Fp(\016)29 b(>)c Fq(0)33 b Fm(with)g Fp(S)1934 4461 y Fn(d;\016)2053 4446 y Fl(\032)25 b Fp(S)2205 4461 y Fn(\016)2275 4446 y Fm(such)33 b(that)h(the)f(formal)h(solution)1411 4725 y Fq(^)-51 b Fp(u)p Fq(\()p Fp(t;)15 b(z)1607 4739 y Fo(1)1647 4725 y Fp(;)g(z)1729 4739 y Fo(2)1769 4725 y Fq(\))26 b(=)1956 4611 y Fd(1)1926 4639 y Fh(X)1930 4834 y Fn(j)t Fo(=0)2082 4663 y Fp(t)2115 4630 y Fn(j)p 2082 4704 70 4 v 2083 4787 a Fp(j)5 b Fq(!)2162 4725 y Fp(u)2214 4739 y Fn(j)2250 4725 y Fq(\()p Fp(z)2327 4739 y Fo(1)2367 4725 y Fp(;)15 b(z)2449 4739 y Fo(2)2489 4725 y Fq(\))75 5024 y Fm(wher)-5 b(e)34 b Fp(u)384 5038 y Fn(j)453 5024 y Fm(ar)-5 b(e)33 b(de\014ne)-5 b(d)34 b(by)e(the)h(formula)h Fq(\()p Fp(u)1608 5038 y Fn(j)1645 5024 y Fq(\))p Fm(,)f(is)g Fp(m)p Fl(\000)p Fm(summable)g(in)f(the)h(dir)-5 b(e)g(ction)34 b Fp(d)p Fm(.)75 5311 y Fr(4)135 b(Multisummabilit)l(y)47 b(of)e(formal)h (solutions)75 5514 y Fq(The)23 b(next)h(result)g(explains)g(ho)m(w)g (to)g(construct)g(linear)h(partial)f(di\013eren)m(tial)i(equations)e (with)g(constan)m(t)h(co)s(ef-)75 5627 y(\014cien)m(ts)j(ha)m(ving)g (solutions)g(whic)m(h)f(are)h(pro)s(ducts)e(of)h(p)s(o)m(w)m(er)h (series)f(solutions)h(of)g(giv)m(en)g(partial)g(di\013eren)m(tial)75 5740 y(equations.)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 3764 100 a Fq(17)75 399 y Ff(Prop)s(osition)36 b(8)46 b Fm(L)-5 b(et)32 b Fp(P)952 413 y Fn(\013)1027 399 y Fl(2)24 b Fe(C)p Fq([)p Fp(t;)15 b(z)1318 413 y Fo(1)1358 399 y Fp(;)g(:)g(:)g(:)i(;)e(z)1602 414 y Fn(d)1643 399 y Fq(])p Fm(,)32 b Fp(P)1786 414 y Fn(\014)1858 399 y Fl(2)25 b Fe(C)p Fq([)p Fp(t;)15 b(z)2150 414 y Fn(d)p 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Fp(;)g(:)g(:)g(:)i(;)e(z)1893 1642 y Fn(p)1933 1628 y Fq(\))24 b Fm(for)g(which)g(the)g(assumption)h Ff(A)f Fm(is)f(satis\014e)-5 b(d)25 b(and)f(such)75 1740 y(that)39 b Fp(P)324 1755 y Fn(\013\014)416 1740 y Fq(\()p Fp(@)499 1754 y Fn(t)530 1740 y Fp(;)15 b(@)618 1754 y Fn(z)651 1763 y Fk(1)690 1740 y Fp(;)g(:)g(:)g(:)i(;)e(@)940 1754 y Fn(z)973 1762 y Fg(p)1013 1740 y Fq(\))6 b(^)-51 b Fp(u)1100 1754 y Fn(\013)1156 1740 y Fq(^)g Fp(u)1202 1755 y Fn(\014)1283 1740 y Fq(=)34 b(0)p Fm(.)56 b(In)38 b(p)-5 b(articular,)40 b(if)j Fq(^)-51 b Fp(u)2235 1754 y Fn(\013)2322 1740 y Fm(\(r)-5 b(esp.)63 b Fq(^)-51 b Fp(u)2654 1755 y Fn(\014)2701 1740 y Fm(\))38 b(is)f Fq(\()p Fp(m)2993 1754 y Fo(1)3033 1740 y Fp(;)15 b(:)g(:)g(:)h(;)f(m) 3314 1754 y Fn(\013)3359 1763 y Fk(1)3399 1740 y Fq(\))p Fm(-summable)75 1853 y(\(r)-5 b(esp.)73 b Fq(\()p Fp(m)486 1820 y Fd(0)486 1878 y Fo(1)525 1853 y Fp(;)15 b(:)g(:)g(:)i(;)e(m)807 1820 y Fd(0)807 1881 y Fn(\014)847 1890 y Fk(1)885 1853 y Fq(\))p 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Fn(p)2126 5347 y Fq(\))p Fl(j)26 b(\024)f Fq(\()p Fp(C)2408 5361 y Fn(\013)2478 5347 y Fq(+)20 b Fp(C)2634 5362 y Fn(\014)2681 5347 y Fq(\))2778 5229 y Fn(p)2738 5261 y Fh(Y)2731 5459 y Fn(k)r Fo(=1)2875 5347 y Fl(j)p Fp(w)2965 5362 y Fn(k)3008 5347 y Fl(j)3033 5310 y Fn(m)3095 5322 y Fg(k)3138 5347 y Fp(;)75 5627 y Fq(for)34 b Fl(j)p Fp(w)308 5642 y Fn(k)351 5627 y Fl(j)e(\025)g Fp(R)581 5594 y Fd(0)580 5652 y Fo(0)651 5627 y Fq(=)g(sup)o(\()p Fp(R)995 5641 y Fo(0)1035 5627 y Fp(;)15 b Fq(1\),)37 b(1)32 b Fl(\024)g Fp(j)37 b Fl(\024)32 b Fp(p)p Fq(.)52 b(So)35 b(that)g(the)f(assumptions)g Ff(A)g Fq(i\),)j(ii\),)f(iii\))f (are)g(satis\014ed)g(for)75 5740 y(the)c(p)s(olynomial)f Fp(P)759 5755 y Fn(\013\014)852 5740 y Fq(.)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 75 100 a Fq(18)216 399 y(In)24 b(the)i(follo)m(wing,)i(w)m(e)d(k)m(eep)h(the)f(same)h(notations)g(as)f (in)g(the)g(section)h(1.)40 b(F)-8 b(or)26 b(all)f(non)g(negativ)m(e)i (in)m(tegers)75 511 y Fp(m)p Fq(,)j Fp(n)265 526 y Fn(k)308 511 y Fq(,)g(1)c Fl(\024)f Fp(k)j Fl(\024)d Fp(p)p Fq(,)31 b(w)m(e)f(ha)m(v)m(e)i(that)75 748 y(\()p Fp(B)5 b Fq(\))516 624 y Fh(I)566 831 y Fn(q)r(C)650 842 y Fg(R)730 685 y Fq(\()p Fp(\025)818 699 y Fn(\013)888 685 y Fq(+)20 b Fp(\025)1032 700 y Fn(\014)1079 685 y Fq(\))1114 652 y Fn(m)p 730 727 452 4 v 734 824 a Fq(\005)802 780 y Fn(p)802 853 y(k)r Fo(=1)935 824 y Fp(w)1002 781 y Fn(n)1045 793 y Fg(k)1083 781 y Fo(+1)1000 853 y Fn(k)1191 748 y Fq(\005)1259 704 y Fn(p)1259 778 y(k)r Fo(=1)1392 748 y Fp(e)1434 711 y Fn(w)1485 723 y Fg(k)1523 711 y Fn(z)1556 723 y Fg(k)1598 748 y Fp(dw)1710 762 y Fo(1)1765 748 y Fl(\001)15 b(\001)g(\001)h Fp(dw)1998 762 y Fn(p)2064 748 y Fq(=)2210 662 y Fh(X)2160 859 y Fn(k)r Fo(+)p Fn(l)q Fo(=)p Fn(m)2430 687 y Fp(m)p Fq(!)p 2417 727 130 4 v 2417 811 a Fp(k)s Fq(!)p Fp(l)r Fq(!)2557 748 y Fp(A)2625 763 y Fn(k)r(;n)2727 772 y Fk(1)2761 763 y Fn(;:::)n(;n)2902 775 y Fg(d)2942 748 y Fp(A)3010 763 y Fn(l)q(;n)3095 775 y Fg(d)p Fk(+1)3207 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Fg(R)1487 1487 y Fp(\025)1540 1454 y Fn(l)1540 1515 y(\014)1587 1487 y Fq(\()p Fp(w)1687 1502 y Fn(d)p Fo(+1)1818 1487 y Fp(;)f(:)g(:)g(:)i(;)e(w)2085 1501 y Fn(p)2125 1487 y Fq(\))p 1487 1542 674 4 v 1556 1639 a(\005)1624 1595 y Fn(p)1624 1669 y(k)r Fo(=)p Fn(d)p Fo(+1)1848 1639 y Fp(w)1915 1596 y Fn(n)1958 1608 y Fg(k)1997 1596 y Fo(+1)1913 1669 y Fn(k)2170 1563 y Fq(\005)2238 1519 y Fn(p)2238 1593 y(k)r Fo(=)p Fn(d)p Fo(+1)2462 1563 y Fp(e)2504 1526 y Fn(w)2555 1538 y Fg(k)2593 1526 y Fn(z)2626 1538 y Fg(k)2668 1563 y Fp(dw)2780 1578 y Fn(d)p Fo(+1)2927 1563 y Fl(\001)g(\001)g(\001)h Fp(dw)3160 1577 y Fn(p)3200 1563 y Fp(:)75 1795 y Fq(Consider)32 b(no)m(w)h(the)g(series)g Fp(u)1106 1762 y Fn(\013)1106 1823 y(k)1188 1795 y Fq(of)g(the)g(prop)s(osition)g(2)g(constructed)g (from)f(the)h(function)g Fp(\025)3293 1809 y Fn(\013)3375 1795 y Fq(and)g(\010)3621 1809 y Fn(\013)3670 1795 y Fq(,)g(the)75 1921 y(series)27 b Fp(u)367 1877 y Fn(\014)367 1951 y(l)442 1921 y Fq(of)g(the)g(prop)s(osition)g(2)h(constructed)f (from)g(the)g(function)g Fp(\025)2501 1936 y Fn(\014)2575 1921 y Fq(and)g(\010)2815 1936 y Fn(\014)2889 1921 y Fq(and)f(the)i(series)f Fp(u)3508 1877 y Fn(\013\014)3508 1933 y(m)3628 1921 y Fq(of)g(the)75 2034 y(prop)s(osition)j(2)g (constructed)h(from)f(the)g(function)g Fp(\025)1899 2048 y Fn(\013)1968 2034 y Fq(+)20 b Fp(\025)2112 2049 y Fn(\014)2189 2034 y Fq(and)30 b(\010)2432 2048 y Fn(\013)2481 2034 y Fq(\010)2547 2049 y Fn(\014)2594 2034 y Fq(.)40 b(F)-8 b(rom)31 b(the)f(equalit)m(y)i(\()p Fp(B)5 b Fq(\),)31 b(w)m(e)f(get)75 2147 y(that)875 2249 y Fp(u)927 2204 y Fn(\013\014)927 2260 y(m)1019 2249 y Fq(\()p Fp(z)1096 2263 y Fo(1)1136 2249 y Fp(;)15 b(:)g(:)g(:)i(;)e(z)1380 2263 y Fn(p)1420 2249 y Fq(\))p 875 2290 581 4 v 1112 2373 a Fp(m)p Fq(!)1491 2311 y(=)1637 2224 y Fh(X)1587 2422 y Fn(k)r Fo(+)p Fn(l)q Fo(=)p Fn(m)1844 2247 y Fp(u)1896 2214 y Fn(\013)1896 2275 y(k)1946 2247 y Fq(\()p Fp(z)2023 2261 y Fo(1)2063 2247 y Fp(;)g(:)g(:)g(:)i(;)e(z)2307 2262 y Fn(d)2348 2247 y Fq(\))p 1844 2290 539 4 v 2076 2373 a Fp(k)s Fq(!)2403 2246 y Fp(u)2455 2202 y Fn(\014)2455 2275 y(l)2502 2246 y Fq(\()p Fp(z)2579 2261 y Fn(d)p Fo(+1)2710 2246 y Fp(;)g(:)g(:)g(:)i(;)e(z)2954 2260 y Fn(p)2994 2246 y Fq(\))p 2403 2290 627 4 v 2689 2373 a Fp(l)r Fq(!)3040 2311 y Fp(:)75 2585 y Fq(W)-8 b(e)35 b(conclude)g(that)g(the)f(formal)g(series)41 b(^)-51 b Fp(u)1563 2600 y Fn(\013\014)1689 2585 y Fq(of)35 b(the)f(prop)s (osition)g(2)g(constructed)h(from)e(the)i(series)f(\()p Fp(u)3726 2540 y Fn(\013\014)3726 2596 y(m)3819 2585 y Fq(\))75 2697 y(satis\014es)j(^)-51 b Fp(u)462 2712 y Fn(\013\014)580 2697 y Fq(=)30 b(^)-50 b Fp(u)728 2711 y Fn(\013)783 2697 y Fq(^)f Fp(u)829 2712 y Fn(\014)876 2697 y Fq(.)41 b(F)-8 b(rom)31 b(the)f(prop)s(osition)g(2)h(it)g(also)g (follo)m(ws)h(that)1418 2890 y Fp(P)1476 2905 y Fn(\013\014)1568 2890 y Fq(\()p Fp(@)1651 2904 y Fn(t)1682 2890 y Fp(;)15 b(@)1770 2904 y Fn(z)1803 2913 y Fk(1)1842 2890 y Fp(;)g(:)g(:)g(:)i(;) e(@)2092 2904 y Fn(z)2125 2912 y Fg(p)2165 2890 y Fq(\))6 b(^)-51 b Fp(u)2252 2905 y Fn(\013\014)2370 2890 y Fq(=)25 b(0)75 3082 y(and)36 b(^)-51 b Fp(u)304 3097 y Fn(\013\014)396 3082 y Fq(\(0)p Fp(;)15 b(z)558 3096 y Fo(1)599 3082 y Fp(;)g(:)g(:)g(:)i(;)e(z)843 3096 y Fn(p)883 3082 y Fq(\))26 b(=)f(\010)1106 3096 y Fn(\013)1155 3082 y Fq(\010)1221 3097 y Fn(\014)1268 3082 y Fq(.)2463 b Fc(2)75 3287 y Ff(Example)35 b(2)30 b Fq(W)-8 b(e)32 b(consider)e(the)h(m)m (ultidimensional)g(heat)g(equation)75 3548 y(\()p Fp(E)5 b Fq(2\))1061 b(\()p Fp(@)1406 3562 y Fn(t)1457 3548 y Fl(\000)1595 3430 y Fn(p)1548 3462 y Fh(X)1549 3660 y Fn(k)r Fo(=1)1694 3548 y Fp(@)1747 3511 y Fo(2)1742 3571 y Fn(z)1775 3583 y Fg(k)1817 3548 y Fq(\))p Fp(u)p Fq(\()p Fp(t;)15 b(z)2054 3562 y Fo(1)2095 3548 y Fp(;)g(:)g(:)g(:)i(;) e(z)2339 3562 y Fn(p)2379 3548 y Fq(\))26 b(=)f(0)p Fp(;)75 3825 y Fq(with)j Fp(u)p Fq(\(0)p Fp(;)15 b(z)494 3839 y Fo(1)535 3825 y Fp(;)g(:)g(:)g(:)i(;)e(z)779 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4299 y Fg(p)p Fj(\000)p Fk(1)2192 4276 y Fq(\))6 b(^)-51 b Fp(u)2279 4290 y Fn(\013)2354 4276 y Fq(=)25 b(0)p Fp(;)75 4452 y Fq(where)38 b Fp(P)404 4466 y Fn(\013)453 4452 y Fq(\()p Fp(t;)15 b(z)603 4466 y Fo(1)644 4452 y Fp(;)g(:)g(:)g(:)h(;)f(z)887 4466 y Fn(p)p Fd(\000)p Fo(1)1018 4452 y Fq(\))38 b(=)g Fp(t)25 b Fl(\000)1355 4384 y Fh(P)1451 4408 y Fn(p)p Fd(\000)p Fo(1)1451 4482 y Fn(k)r Fo(=1)1599 4452 y Fp(z)1645 4419 y Fo(2)1641 4480 y Fn(k)1684 4452 y Fq(,)40 b(with)k(^)-51 b Fp(u)2016 4466 y Fn(\013)2066 4452 y Fq(\(0)p Fp(;)15 b(z)2228 4466 y Fo(1)2269 4452 y Fp(;)g(:)g(:)g(:)h(;)f(z)2512 4466 y Fn(p)p Fd(\000)p Fo(1)2643 4452 y Fq(\))38 b(=)g(\010)2891 4466 y Fo(1)2930 4452 y Fq(\()p Fp(z)3007 4466 y Fo(1)3047 4452 y Fq(\))15 b Fl(\001)g(\001)g(\001)i Fq(\010)3285 4466 y Fn(p)p Fd(\000)p Fo(1)3415 4452 y Fq(\()p Fp(z)3492 4466 y Fn(p)p Fd(\000)p Fo(1)3622 4452 y Fq(\),)41 b(has)75 4565 y(a)36 b(formal)f(series)41 b(^)-51 b Fp(u)746 4579 y Fn(\013)831 4565 y Fq(solution)36 b(\(2\))g(whic)m(h)f(is)g (1-summable)g(with)g(resp)s(ect)g(to)h Fp(t)f Fq(in)g(all)h(but)e (\014nitely)h(man)m(y)75 4678 y(directions.)41 b(In)30 b(this)g(case)h(the)g(ro)s(ot)f(of)h Fp(P)1521 4692 y Fn(\013)1601 4678 y Fq(in)e(the)i(v)-5 b(ariable)31 b Fp(t)f Fq(is)g(the)h(function)f Fp(\025)2924 4692 y Fn(\013)2998 4678 y Fq(=)3094 4610 y Fh(P)3190 4636 y Fn(d)p Fd(\000)p Fo(1)3190 4705 y Fn(k)r Fo(=1)3338 4678 y Fp(z)3384 4645 y Fo(2)3380 4706 y Fn(k)3424 4678 y Fq(.)41 b(F)-8 b(rom)30 b(the)75 4791 y(pap)s(ers)f([2],)j([6],)f(w)m(e)g(kno)m(w)f(that)h (heat)g(equation)1610 4983 y Fp(P)1668 4998 y Fn(\014)1716 4983 y Fq(\()p Fp(@)1799 4997 y Fn(t)1829 4983 y Fp(;)15 b(@)1917 4997 y Fn(z)1950 5005 y Fg(p)1991 4983 y Fq(\))6 b(^)-51 b Fp(u)2078 4997 y Fn(\013)2153 4983 y Fq(=)25 b(0)p Fp(;)75 5176 y Fq(where)34 b Fp(P)400 5191 y Fn(\014)448 5176 y Fq(\()p Fp(t;)15 b(z)598 5190 y Fn(p)638 5176 y Fq(\))33 b(=)g Fp(t)23 b Fl(\000)g Fp(z)1006 5143 y Fo(2)1002 5198 y Fn(p)1045 5176 y Fq(,)36 b(with)41 b(^)-51 b Fp(u)1370 5191 y Fn(\014)1417 5176 y Fq(\(0)p Fp(;)15 b(z)1579 5190 y Fn(p)1620 5176 y Fq(\))33 b(=)g(\010)1858 5190 y Fn(p)1897 5176 y Fq(\()p Fp(z)1974 5190 y Fn(p)2014 5176 y Fq(\),)k(has)d(a)h(formal)h(series)k(^)-50 b Fp(u)2948 5191 y Fn(\014)3030 5176 y Fq(solution)35 b(\(2\))h(whic)m(h)e(is)75 5289 y(1-summable)c(with)f(resp)s(ect)g(to)h Fp(t)g Fq(in)f(all)h(but)f (\014nitely)g(man)m(y)h(directions.)41 b(The)29 b(ro)s(ot)g(of)h Fp(P)3206 5304 y Fn(\014)3283 5289 y Fq(in)f(the)h(v)-5 b(ariable)75 5402 y Fp(t)33 b Fq(is)g(the)h(function)e Fp(\025)806 5417 y Fn(\014)884 5402 y Fq(=)d Fp(z)1030 5369 y Fo(2)1026 5424 y Fn(p)1070 5402 y Fq(.)49 b(On)32 b(the)i(other)f(hand,)g(the)h(p)s(olynomial)f Fp(P)2644 5417 y Fn(\013\014)2737 5402 y Fq(\()p Fp(t;)15 b(z)2887 5416 y Fo(1)2927 5402 y Fp(;)g(:)g(:)g(:)i(;)e(z)3171 5416 y Fn(p)3211 5402 y Fq(\))31 b(=)e Fp(t)22 b Fl(\000)3525 5333 y Fh(P)3621 5357 y Fn(p)3621 5431 y(k)r Fo(=1)3769 5402 y Fp(z)3815 5369 y Fo(2)3811 5429 y Fn(k)75 5514 y Fq(satisfy)31 b Fp(P)417 5529 y Fn(\013\014)509 5514 y Fq(\()p Fp(\025)597 5528 y Fn(\013)667 5514 y Fq(+)20 b Fp(\025)811 5529 y Fn(\014)858 5514 y Fp(;)15 b(z)940 5528 y Fo(1)980 5514 y Fp(;)g(:)g(:)g(:)i(;)e(z)1224 5528 y Fn(p)1264 5514 y Fq(\))26 b(=)f(0.)41 b(F)-8 b(rom)30 b(the)h(prop)s(osition)f(8,)g(w)m(e)h(deduce)f(that)h(the)f(equation)h (\()p Fp(E)5 b Fq(2\))75 5627 y(has)30 b(the)g(formal)h(series)36 b(^)-51 b Fp(u)974 5641 y Fn(\013)1029 5627 y Fq(^)g Fp(u)1075 5642 y Fn(\014)1153 5627 y Fq(as)30 b(solution,)h(whic)m(h)f (is)g(1-summable)g(with)g(resp)s(ect)g(to)h Fp(t)f Fq(in)g(all)h(but)e (\014nitely)75 5740 y(man)m(y)h(directions.)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 3764 100 a Fq(19)75 399 y Ff(Example)35 b(3)30 b Fq(Consider)g(the)g(partial)i(di\013eren)m(tial)f(equation)75 603 y(\()p Fp(E)5 b Fq(3\))1102 b(\()p Fp(@)1447 617 y Fn(t)1497 603 y Fl(\000)20 b Fp(@)1641 565 y Fo(2)1636 625 y Fn(z)1669 634 y Fk(1)1728 603 y Fl(\000)g Fp(@)1872 565 y Fo(3)1867 625 y Fn(z)1900 634 y Fk(2)1939 603 y Fq(\))p Fp(u)p Fq(\()p Fp(t;)15 b(z)2176 617 y Fo(1)2216 603 y Fp(;)g(z)2298 617 y Fo(2)2339 603 y Fq(\))25 b(=)g(0)p Fp(;)75 807 y Fq(with)32 b Fp(u)p Fq(\(0)p Fp(;)15 b(z)498 821 y Fo(1)539 807 y Fp(;)g(z)621 821 y Fo(2)661 807 y Fq(\))29 b(=)f(\010)890 821 y Fo(1)929 807 y Fq(\()p Fp(z)1006 821 y Fo(1)1046 807 y Fq(\)\010)1147 821 y Fo(2)1186 807 y Fq(\()p Fp(z)1263 821 y Fo(2)1303 807 y Fq(\),)34 b(where)d(\010)1727 821 y Fo(1)1798 807 y Fq(and)h(\010)2043 821 y Fo(2)2114 807 y Fq(are)h(rational)g (functions.)46 b(F)-8 b(rom)33 b(the)f(pap)s(ers)f([2],)75 920 y([6],)h(w)m(e)e(kno)m(w)h(that)g(the)f(equation)1610 1033 y Fp(P)1668 1047 y Fn(\013)1718 1033 y Fq(\()p Fp(@)1801 1047 y Fn(t)1831 1033 y Fp(;)15 b(@)1919 1047 y Fn(z)1952 1056 y Fk(1)1991 1033 y Fq(\))6 b(^)-51 b Fp(u)2078 1047 y Fn(\013)2153 1033 y Fq(=)25 b(0)p Fp(;)75 1200 y Fq(where)34 b Fp(P)400 1214 y Fn(\013)450 1200 y Fq(\()p Fp(t;)15 b(z)600 1214 y Fo(1)640 1200 y Fq(\))33 b(=)f Fp(t)22 b Fl(\000)h Fp(z)1006 1167 y Fo(2)1002 1224 y(1)1046 1200 y Fq(,)35 b(with)40 b(^)-50 b Fp(u)1370 1214 y Fn(\013)1419 1200 y Fq(\(0)p Fp(;)15 b(z)1581 1214 y Fo(1)1622 1200 y Fq(\))33 b(=)f(\010)1859 1214 y Fo(1)1898 1200 y Fq(\()p Fp(z)1975 1214 y Fo(1)2015 1200 y Fq(\),)k(has)e(a)h(formal)g(series)40 b(^)-50 b Fp(u)2947 1214 y Fn(\013)3030 1200 y Fq(solution)36 b(\(2\))f(whic)m(h)f(is)75 1313 y(1-summable)k(with)f(resp)s(ect)g(to)h Fp(t)f Fq(in)g(all)i(but)d(\014nitely)i(man)m(y)f(directions.)63 b(In)36 b(this)h(case)i(the)e(ro)s(ot)h(of)g Fp(P)3805 1327 y Fn(\013)75 1425 y Fq(in)c(the)g(v)-5 b(ariable)35 b Fp(t)e Fq(is)h(the)g(function)g Fp(\025)1424 1439 y Fn(\013)1505 1425 y Fq(=)d Fp(z)1653 1392 y Fo(2)1649 1450 y(1)1692 1425 y Fq(.)52 b(Lik)m(ewise,)36 b(from)d(the)i(pap)s (ers)d([2],)k([7],)g(w)m(e)f(kno)m(w)f(that)g(the)75 1538 y(equation)1612 1651 y Fp(P)1670 1666 y Fn(\014)1718 1651 y Fq(\()p Fp(@)1801 1665 y Fn(t)1831 1651 y Fp(;)15 b(@)1919 1665 y Fn(z)1952 1674 y Fk(2)1991 1651 y Fq(\))6 b(^)-51 b Fp(u)2078 1666 y Fn(\014)2151 1651 y Fq(=)25 b(0)p Fp(;)75 1818 y Fq(where)f Fp(P)390 1833 y Fn(\014)437 1818 y Fq(\()p Fp(t;)15 b(z)587 1832 y Fo(2)627 1818 y Fq(\))26 b(=)f Fp(t)8 b Fl(\000)g Fp(z)950 1785 y Fo(3)946 1843 y(2)988 1818 y Fq(,)26 b(with)j(^)-50 b Fp(u)1292 1833 y Fn(\014)1339 1818 y Fq(\(0)p Fp(;)15 b(z)1501 1832 y Fo(2)1542 1818 y Fq(\))25 b(=)g(\010)1764 1832 y Fo(2)1803 1818 y Fq(\()p Fp(z)1880 1832 y Fo(2)1920 1818 y Fq(\),)h(has)e(a)g(formal)h(series)30 b(^)-51 b Fp(u)2800 1833 y Fn(\014)2871 1818 y Fq(solution)25 b(\(2\))g(whic)m(h)f(is)g(1/2-)75 1931 y(summable)j(with)g(resp)s(ect)h (to)g Fp(t)f Fq(in)h(all)g(but)f(\014nitely)h(man)m(y)f(directions.)41 b(In)27 b(this)g(case)i(the)e(ro)s(ot)h(of)g Fp(P)3550 1946 y Fn(\014)3625 1931 y Fq(in)f(the)75 2044 y(v)-5 b(ariable)27 b Fp(t)g Fq(is)f(the)h(function)f Fp(\025)1117 2059 y Fn(\014)1190 2044 y Fq(=)f Fp(z)1332 2011 y Fo(3)1328 2068 y(2)1372 2044 y Fq(.)39 b(On)26 b(the)g(other)h(hand,)g(the)g(p)s (olynomial)f Fp(P)2896 2059 y Fn(\013\014)2989 2044 y Fq(\()p Fp(t;)15 b(z)3139 2058 y Fo(1)3180 2044 y Fp(;)g(z)3262 2058 y Fo(2)3302 2044 y Fq(\))25 b(=)g Fp(t)13 b Fl(\000)g Fp(z)3634 2011 y Fo(2)3630 2068 y(1)3685 2044 y Fl(\000)g Fp(z)3815 2011 y Fo(3)3811 2068 y(2)75 2157 y Fq(satisfy)31 b Fp(P)417 2172 y Fn(\013\014)509 2157 y Fq(\()p Fp(\025)597 2171 y Fn(\013)667 2157 y Fq(+)20 b Fp(\025)811 2172 y Fn(\014)858 2157 y Fp(;)15 b(z)940 2171 y Fo(1)980 2157 y Fp(;)g(z)1062 2171 y Fo(2)1102 2157 y Fq(\))26 b(=)f(0.)41 b(F)-8 b(rom)30 b(the)h(prop)s(osition)f(8,)g(w)m(e)h (deduce)f(that)h(the)f(equation)h(\()p Fp(E)5 b Fq(3\))32 b(has)75 2270 y(the)25 b(formal)g(series)31 b(^)-51 b Fp(u)796 2284 y Fn(\013)851 2270 y Fq(^)g Fp(u)897 2285 y Fn(\014)969 2270 y Fq(as)25 b(solution,)i(whic)m(h)d(is)h(\(1)p Fp(;)15 b Fq(1)p Fp(=)p Fq(2\)-summable)27 b(with)e(resp)s(ect)f(to)i Fp(t)e Fq(in)h(all)g(but)f(\014nitely)75 2383 y(man)m(y)j(directions,)i (so)e(that)g(the)h(series)33 b(^)-51 b Fp(u)1504 2397 y Fn(\013)1559 2383 y Fq(^)h Fp(u)1606 2398 y Fn(\014)1680 2383 y Fq(is)27 b(not)g Fp(m)p Fq(-summable)g(in)f(an)m(y)i(direction)f Fp(d)p Fq(,)i(for)d(an)m(y)i Fp(m)d Fl(2)g Fe(R)3771 2397 y Fo(+)3829 2383 y Fq(.)75 2669 y Fr(References)75 2872 y Fq([1])97 b(W.)21 b(Balser,)j Fm(Multisummability)g(of)g(formal) h(p)-5 b(ower)25 b(series)f(solutions)h(of)f(p)-5 b(artial)26 b(di\013er)-5 b(ential)24 b(e)-5 b(quations)267 2985 y(with)33 b(c)-5 b(onstant)35 b(c)-5 b(o)g(e\016cients)p Fq(,)31 b(preprin)m(t,)f(submitted)g(to)h(Elsevier)g(Science)g (\(2002\).)75 3173 y([2])97 b(W.)22 b(Balser,)i Fm(Summability)i(of)e (formal)i(p)-5 b(ower)26 b(series)f(solutions)g(of)g(p)-5 b(artial)27 b(di\013er)-5 b(ential)25 b(e)-5 b(quations)26 b(with)267 3286 y(c)-5 b(onstant)29 b(c)-5 b(o)g(e\016cients)p Fq(,)27 b(preprin)m(t,)f(submitted)e(to)h(Pro)s(ceedings)g(of)g(the)h (In)m(ternational)g(Conference)f(on)267 3398 y(Di\013eren)m(tial)33 b(and)c(F)-8 b(unctional)32 b(Di\013eren)m(tial)h(Equations,)e(Mosco)m (w)h(\(2002\).)75 3586 y([3])97 b(W.)39 b(Balser,)j Fm(F)-7 b(ormal)42 b(p)-5 b(ower)42 b(series)e(and)h(line)-5 b(ar)41 b(systems)g(of)g(mer)-5 b(omorphic)43 b(or)-5 b(dinary)42 b(di\013er)-5 b(ential)267 3699 y(e)g(quations)p Fq(,)32 b(Springer-V)-8 b(erlag,)31 b(New-Y)-8 b(ork,)32 b(2000.)75 3887 y([4])97 b(Jarosla)m(w)29 b(Cel,)g Fm(The)i (Newton-Puiseux)f(the)-5 b(or)g(em)33 b(for)f(sever)-5 b(al)31 b(variables)p Fq(,)f(Bull.)f(So)s(c.)g(Sci.)f(Lett.)i(Lo)s(dz) 267 4000 y(40)h(\(1990\),)i(no.)e(11-20,)h(p.)e(53{61.)75 4187 y([5])97 b(L.)41 b(H\177)-45 b(ormander,)42 b Fm(A)n(n)g(intr)-5 b(o)g(duction)44 b(to)f(c)-5 b(omplex)43 b(analysis)h(in)e(sever)-5 b(al)43 b(variables)p Fq(,)h(second)d(edition,)267 4300 y(North-Holland,)31 b(1973.)75 4488 y([6])97 b(D.)27 b(A.)g(Lutz,)g(M.)g(Miy)m(ak)m(e,)j(R.)d(Sc)m(haefk)m(e,)i Fm(On)g(the)g(Bor)-5 b(el)30 b(summability)g(of)g(diver)-5 b(gent)29 b(solutions)i(of)e(the)267 4601 y(he)-5 b(at)34 b(e)-5 b(quation)p Fq(,)31 b(Nago)m(y)m(a)i(Math.)f(J.)e(154)i (\(1999\))g(53{63.)75 4788 y([7])97 b(M.)54 b(Miy)m(ak)m(e,)62 b Fm(Bor)-5 b(el)55 b(summability)g(of)g(diver)-5 b(gent)54 b(solutions)i(of)e(the)g(Cauchy)h(pr)-5 b(oblem)56 b(to)e(non-)267 4901 y(Kovaleskian)41 b(e)-5 b(quations)p Fq(,)41 b(in:)55 b(C.)38 b(Hua,)i(L.)e(Ro)s(dino)g(\(Eds.\),)i(P)m(artial)g(Di\013eren)m (tial)g(Equations)e(and)267 5014 y(Their)30 b(Applications,)h(W)-8 b(orld)31 b(Scien)m(ti\014c,)h(Singap)s(ore,)e(1999,)i(pp.)e(225{239.) 75 5202 y([8])97 b(S.)26 b(Ouc)m(hi,)h Fm(Multisummability)j(of)f (formal)i(solutions)f(of)f(some)h(line)-5 b(ar)30 b(p)-5 b(artial)31 b(di\013er)-5 b(ential)30 b(e)-5 b(quations)p Fq(,)267 5315 y(J.)30 b(Di\013eren)m(tial)j(Equations)d(185)i (\(2002\),)h(no.)e(2,)g(513{549.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0409230749612--