Content-Type: multipart/mixed; boundary="-------------0603130941535" This is a multi-part message in MIME format. ---------------0603130941535 Content-Type: text/plain; name="06-66.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-66.keywords" asymptotic expansion, Cauchy problem, partial differential equation, Gevrey estimates, power series solution, Borel summability, shrinking, deviating variable ---------------0603130941535 Content-Type: application/postscript; name="nlpddiff.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nlpddiff.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: nlpddiff.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 CMBX12 CMMI10 CMMI8 CMR6 CMR8 %%+ CMSY8 CMSY10 MSBM10 CMEX10 CMTI10 CMMI6 CMSY6 LASY10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o nlpddiff.ps nlpddiff.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2006.03.13:1126 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % 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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 11 /ff put dup 12 /fi put dup 19 /acute put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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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 12 /fi put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 75 /K put dup 76 /L put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec 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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 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt 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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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 readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 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 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 14 /openbullet put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 68 /D put dup 73 /I put dup 77 /M put dup 80 /P put dup 83 /S put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 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 /minus put dup 21 /greaterequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 68 /D put dup 73 /I put dup 77 /M put dup 83 /S put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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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. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 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 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 61 /equal put dup 67 /C put dup 97 /a put dup 100 /d put dup 114 /r put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 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 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % 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. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 22 /mu put dup 23 /nu put dup 26 /rho put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 68 /D put dup 75 /K put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 100 /d put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 116 /t put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 88 /X put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 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 /Gamma put dup 5 /Pi put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % 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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 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 11 /ff put dup 45 /hyphen put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 65 /A put dup 70 /F put dup 73 /I put dup 82 /R put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 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 19 /acute put dup 39 /quoteright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 57 /nine put dup 64 /at put dup 65 /A put dup 67 /C put dup 70 /F put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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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b(of)i(asymptotic)g(analysis)f (of)g(solutions)h(to)75 3347 y(functional)e(ordinary)e(or)i(partial)g (di\013eren)m(tial)g(equations.)48 b(F)-8 b(or)33 b(recen)m(t)g (results)f(in)g(this)g(\014eld,)h(w)m(e)g(refer)f(to)75 3460 y([2)q(],[7)q(],[8)q(],[9)q(],[11)r(],[16)r(].)75 3747 y Fo(2)135 b(Summabilit)l(y)46 b(criterions)75 3950 y Fn(In)38 b(this)h(section,)j(w)m(e)e(recall)g(a)f(prop)s(osition)g (already)g(obtained)g(in)g(a)g(previous)g(pap)s(er)e([5)q(],)k(whic)m (h)e(giv)m(es)75 4063 y(necessary)25 b(and)f(su\016cien)m(t)h (conditions)g(for)g(the)f(summabilit)m(y)h(of)g(formal)g(series)g(with) f(v)-5 b(ariable)25 b(co)s(e\016cien)m(ts.)216 4176 y(W)-8 b(e)34 b(consider)f(a)h(sequence)f(\()p Fm(X)1300 4190 y Fl(j)1337 4176 y Fn(\))1372 4190 y Fl(j)t Fi(\025)p Fj(0)1532 4176 y Fn(of)g(holomorphic)g(functions)g Fm(X)2631 4190 y Fl(j)2697 4176 y Fn(:)d Fh(D)2822 4190 y Fl(\032)2892 4176 y Fh(\000)-15 b(!)29 b Fg(C)k Fn(where)f Fh(D)3502 4190 y Fl(\032)3572 4176 y Fn(=)e Fh(f)p Fm(z)k 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Fn(\))26 b(=)f Fm(A)p Fn(\()p Fm(t;)15 b(z)t(;)g Fn(\(\()p Fm(@)1797 5653 y Fl(k)1834 5662 y Fk(0)1792 5718 y Fl(t)1875 5694 y Fm(@)1928 5656 y Fl(k)1965 5665 y Fk(1)1923 5716 y Fl(z)2030 5671 y Fn(^)2004 5694 y Fm(X)7 b Fn(\)\()p Fm(\036)2210 5709 y Fl(k)2247 5718 y Fk(0)2283 5709 y Fl(;k)2340 5718 y Fk(1)2378 5694 y Fn(\()p Fm(t)p Fn(\))p Fm(;)15 b(z)t Fn(\)\))2637 5713 y Fj(\()p Fl(k)2701 5722 y Fk(0)2737 5713 y Fl(;k)2794 5722 y Fk(1)2828 5713 y Fj(\))p Fi(2S)2955 5694 y Fn(\))p Fm(;)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Fn(4)75 399 y(where)27 b Fh(S)34 b Fn(is)28 b(a)g(\014nite)g(subset)f(of)g Fg(N)1258 366 y Fj(2)1297 399 y Fn(.)40 b(The)27 b(function)g Fm(A)p Fn(\()p Fm(t;)15 b(z)t(;)g Fn(\()p Fm(z)2238 414 y Fl(k)2275 423 y Fk(0)2312 414 y Fl(;k)2369 423 y Fk(1)2407 399 y Fn(\))2442 417 y Fj(\()p Fl(k)2506 426 y Fk(0)2541 417 y Fl(;k)2598 426 y Fk(1)2632 417 y Fj(\))p Fi(2S)2758 399 y Fn(\))28 b(is)g(assumed)e(to)j(b)s(e)e(holomor-)75 511 y(phic)k(in)g(the)h(v)-5 b(ariables)33 b(\()p Fm(t;)15 b(z)t Fn(\))28 b Fh(2)f Fm(D)1297 530 y Fj(2)p Fl(R)p Fj(\()p Fl(\032)p Fj(\))1502 511 y Fh(\002)21 b Fm(D)1669 525 y Fl(\032)1741 511 y Fn(and)31 b(p)s(olynomial)h(in)f(the)h(v)-5 b(ariables)32 b(\()p Fm(z)3110 526 y Fl(k)3147 535 y Fk(0)3182 526 y Fl(;k)3239 535 y Fk(1)3277 511 y Fn(\))3312 530 y Fj(\()p Fl(k)3376 539 y Fk(0)3411 530 y Fl(;k)3468 539 y Fk(1)3502 530 y Fj(\))p Fi(2S)3629 511 y Fn(.)44 b(The)75 624 y(function)30 b Fm(A)h Fn(has)f(an)g(expansion)g(of)h(the) f(form)482 837 y Fm(A)p Fn(\()p Fm(t;)15 b(z)t(;)g Fn(\()p Fm(z)821 852 y Fl(k)858 861 y Fk(0)895 852 y Fl(;k)952 861 y Fk(1)990 837 y Fn(\))1025 856 y Fj(\()p Fl(k)1089 865 y Fk(0)1124 856 y Fl(;k)1181 865 y Fk(1)1215 856 y Fj(\))p Fi(2S)1341 837 y Fn(\))26 b(=)1498 751 y Ff(X)1504 948 y Fi(M)1588 957 y Fk(1)1644 837 y Fm(b)1683 856 y Fl(T)1724 865 y Fk(1)1759 856 y Fl(;)p Fj(\()p Fl(T)1847 868 y Fc(k)1880 883 y Fk(0)1914 868 y Fc(;k)1966 883 y Fk(1)2005 856 y Fj(\))2032 870 y Fk(\()p Fc(k)2089 885 y Fk(0)2124 870 y Fc(;k)2176 885 y Fk(1)2210 870 y(\))p Fb(2S)2326 837 y Fn(\()p Fm(t)p Fn(\))p Fm(z)2475 800 y Fl(T)2516 809 y Fk(1)2555 837 y Fn(\005)2623 856 y Fj(\()p Fl(k)2687 865 y Fk(0)2722 856 y Fl(;k)2779 865 y Fk(1)2813 856 y Fj(\))p Fi(2S)2939 837 y Fn(\()p Fm(z)3016 852 y Fl(k)3053 861 y Fk(0)3088 852 y Fl(;k)3145 861 y Fk(1)3183 837 y Fn(\))3218 800 y Fl(T)3259 812 y Fc(k)3292 827 y Fk(0)3327 812 y Fc(;k)3379 827 y Fk(1)3422 837 y Fm(;)75 1129 y Fn(where)1188 1242 y Fh(M)1297 1256 y Fj(1)1361 1242 y Fn(=)f Fh(f)p Fn(\()p Fm(T)1590 1256 y Fj(1)1631 1242 y Fm(;)15 b Fn(\()p Fm(T)1759 1257 y Fl(k)1796 1266 y Fk(0)1831 1257 y Fl(;k)1888 1266 y Fk(1)1926 1242 y Fn(\))1961 1260 y Fj(\()p Fl(k)2025 1269 y Fk(0)2060 1260 y Fl(;k)2117 1269 y Fk(1)2151 1260 y Fj(\))p Fi(2S)2277 1242 y Fn(\))26 b Fh(2)f Fg(N)20 b Fh(\002)g(P)7 b(g)p Fm(;)75 1422 y Fn(where)36 b Fh(P)44 b Fn(is)36 b(a)h(\014nite)f (subset)g(of)h Fg(N)1329 1389 y Fj(Card)o(\()p Fi(S)5 b Fj(\))1588 1422 y Fn(.)59 b(The)36 b(functions)f Fm(\036)2316 1437 y Fl(k)2353 1446 y Fk(0)2388 1437 y Fl(;k)2445 1446 y Fk(1)2483 1422 y Fn(\()p Fm(t)p Fn(\))i(are)g(holomorphic)f(on)g(a)h (common)75 1535 y(neigh)m(b)s(orho)s(o)s(d)29 b(of)h(the)h(origin)g(in) f Fg(C)g Fn(suc)m(h)g(that)h Fm(\036)1822 1550 y Fl(k)1859 1559 y Fk(0)1893 1550 y Fl(;k)1950 1559 y Fk(1)1989 1535 y Fn(\(0\))26 b(=)f(0.)216 1697 y(F)-8 b(rom)31 b(the)g(Cauc)m(h)m(y)f (form)m(ula,)h(w)m(e)g(kno)m(w)f(that)h(for)g(all)g Fm(\027)2175 1711 y Fj(1)2214 1697 y Fm(;)15 b(T)2307 1711 y Fj(1)2347 1697 y Fm(;)g Fn(\()p Fm(T)2475 1712 y Fl(k)2512 1721 y Fk(0)2547 1712 y Fl(;k)2604 1721 y Fk(1)2642 1697 y Fn(\))2677 1716 y Fj(\()p Fl(k)2741 1725 y Fk(0)2777 1716 y Fl(;k)2834 1725 y Fk(1)2867 1716 y Fj(\))p Fi(2S)3019 1697 y Fh(\025)25 b Fn(0,)31 b(w)m(e)g(get)75 1985 y(\()p Fm(C)7 b(I)g Fn(\))644 b(sup)846 2072 y Fl(t)p Fi(2)935 2056 y Fj(\026)918 2072 y Fl(D)976 2086 y Fc(R)p Fk(\()p Fc(\032)p Fk(\))1132 1901 y Fh(j)p Fm(@)1210 1861 y Fl(\027)1245 1870 y Fk(1)1205 1925 y Fl(t)1284 1901 y Fm(b)1323 1920 y Fl(T)1364 1929 y Fk(1)1399 1920 y Fl(;)p Fj(\()p Fl(T)1487 1932 y Fc(k)1520 1947 y Fk(0)1554 1932 y Fc(;k)1606 1947 y Fk(1)1645 1920 y Fj(\))1672 1934 y Fk(\()p Fc(k)1729 1949 y Fk(0)1764 1934 y Fc(;k)1816 1949 y Fk(1)1850 1934 y(\))1878 1920 y Fi(2S)1977 1901 y Fn(\()p Fm(t)p Fn(\))p Fh(j)p 1132 1964 974 4 v 1529 2047 a Fn(\()p Fm(\027)1609 2061 y Fj(1)1648 2047 y Fn(!\))2141 1985 y(=)25 b Fh(j)p Fm(b)p Fh(j)2326 2003 y Fl(T)2367 2012 y Fk(1)2402 2003 y Fl(;)p Fj(\()p Fl(T)2490 2015 y Fc(k)2523 2030 y Fk(0)2558 2015 y Fc(;k)2610 2030 y Fk(1)2648 2003 y Fj(\))2675 2017 y Fk(\()p Fc(k)2732 2032 y Fk(0)2767 2017 y Fc(;k)2819 2032 y Fk(1)2854 2017 y(\))p Fb(2S)2965 2003 y Fl(;\027)3020 2012 y Fk(1)3058 1985 y Fm(;)75 2271 y Fn(with)30 b(the)h(prop)s(ert)m (y)e(that)i(the)g(functions)423 2484 y Fm(A)491 2498 y Fl(K)559 2484 y Fn(\()p Fm(t;)15 b(z)t(;)g Fn(\()p Fm(z)830 2499 y Fl(k)867 2508 y Fk(0)904 2499 y Fl(;k)961 2508 y Fk(1)999 2484 y Fn(\))1034 2502 y Fj(\()p Fl(k)1098 2511 y Fk(0)1133 2502 y Fl(;k)1190 2511 y Fk(1)1224 2502 y Fj(\))p Fi(2S)1350 2484 y Fn(\))26 b(=)1506 2398 y Ff(X)1513 2594 y Fi(M)1597 2603 y Fk(1)1653 2484 y Fh(j)p Fm(b)p Fh(j)1742 2502 y Fl(T)1783 2511 y Fk(1)1818 2502 y Fl(;)p Fj(\()p Fl(T)1906 2514 y Fc(k)1939 2529 y Fk(0)1974 2514 y Fc(;k)2026 2529 y Fk(1)2064 2502 y Fj(\))2091 2516 y Fk(\()p Fc(k)2148 2531 y Fk(0)2183 2516 y Fc(;k)2235 2531 y Fk(1)2270 2516 y(\))p Fb(2S)2385 2484 y Fn(\()p Fm(t)p Fn(\))p Fm(z)2534 2446 y Fl(T)2575 2455 y Fk(1)2615 2484 y Fn(\005)2683 2502 y Fj(\()p Fl(k)2747 2511 y Fk(0)2782 2502 y Fl(;k)2839 2511 y Fk(1)2873 2502 y Fj(\))p Fi(2S)2999 2484 y Fn(\()p Fm(z)3076 2499 y Fl(k)3113 2508 y Fk(0)3148 2499 y Fl(;k)3205 2508 y Fk(1)3243 2484 y Fn(\))3278 2446 y Fl(T)3319 2458 y Fc(k)3352 2473 y Fk(0)3387 2458 y Fc(;k)3439 2473 y Fk(1)3482 2484 y Fm(;)75 2776 y Fn(where)923 2888 y Fh(j)p Fm(b)p Fh(j)1012 2907 y Fl(T)1053 2916 y Fk(1)1088 2907 y Fl(;)p Fj(\()p Fl(T)1176 2919 y Fc(k)1209 2934 y Fk(0)1244 2919 y Fc(;k)1296 2934 y Fk(1)1334 2907 y Fj(\))1361 2921 y Fk(\()p Fc(k)1418 2936 y Fk(0)1453 2921 y Fc(;k)1505 2936 y Fk(1)1540 2921 y(\))p Fb(2S)1655 2888 y Fn(\()p Fm(t)p Fn(\))g(=)1894 2802 y Ff(X)1880 2998 y Fl(\027)1915 3007 y Fk(1)1949 2998 y Fi(\025)p Fj(0)2054 2888 y Fh(j)p Fm(b)p Fh(j)2143 2907 y Fl(T)2184 2916 y Fk(1)2219 2907 y Fl(;)p Fj(\()p Fl(T)2307 2919 y Fc(k)2340 2934 y Fk(0)2375 2919 y Fc(;k)2427 2934 y Fk(1)2466 2907 y Fj(\))2493 2921 y Fk(\()p Fc(k)2550 2936 y Fk(0)2585 2921 y Fc(;k)2637 2936 y Fk(1)2671 2921 y(\))p Fb(2S)2782 2907 y Fl(;\027)2837 2916 y Fk(1)2875 2888 y Fm(t)2908 2851 y Fl(\027)2943 2860 y Fk(1)2981 2888 y Fm(;)75 3144 y Fn(are)42 b(holomorphic)g(in)f(the)h(v)-5 b(ariables)42 b Fm(t;)15 b(z)46 b Fn(in)41 b(a)h(neigh)m(b)s(orho)s(o)s (d)e(of)i(the)g(origin)g(and)f(p)s(olynomial)h(in)f(the)75 3257 y(v)-5 b(ariables)31 b(\()p Fm(z)529 3272 y Fl(k)566 3281 y Fk(0)601 3272 y Fl(;k)658 3281 y Fk(1)696 3257 y Fn(\))731 3275 y Fj(\()p Fl(k)795 3284 y Fk(0)830 3275 y Fl(;k)887 3284 y Fk(1)921 3275 y Fj(\))p Fi(2S)1047 3257 y Fn(.)41 b(W)-8 b(e)32 b(also)f(de\014ne)f(the)g(functions,)1516 3487 y Fm(R)1585 3502 y Fl(k)1622 3511 y Fk(0)1656 3502 y Fl(;k)1713 3511 y Fk(1)1751 3487 y Fn(\()p Fm(t)p Fn(\))c(=)1976 3400 y Ff(X)1986 3598 y Fl(l)q Fi(\025)p Fj(0)2122 3487 y Fm(r)2166 3449 y Fl(l)2163 3510 y(k)2200 3519 y Fk(0)2235 3510 y Fl(;k)2292 3519 y Fk(1)2330 3487 y Fm(t)2363 3449 y Fl(l)2389 3487 y Fm(;)75 3781 y Fn(where)k Fm(r)379 3796 y Fl(k)416 3805 y Fk(0)450 3796 y Fl(;k)507 3805 y Fk(1)576 3781 y Fn(are)g(p)s(ositiv)m(e)i(real)f(n)m(um)m(b)s(ers)e (for)h(all)h Fm(k)1920 3795 y Fj(0)1960 3781 y Fm(;)15 b(k)2047 3795 y Fj(1)2112 3781 y Fh(\025)25 b Fn(0.)216 3944 y(In)40 b(the)g(next)g(lemma)h(w)m(e)g(describ)s(e)e(formal)i (series)f(solutions)h(of)g(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\))40 b(in)g(term)h(of)f(di\013eren)m(tial)75 4057 y(relations.)75 4244 y Fe(Lemma)35 b(1)45 b Fd(R)-5 b(ewrite)34 b(the)f(formal)h(series)f Fn(\()1636 4221 y(^)1610 4244 y Fm(X)8 b Fn(\))33 b Fd(in)f(a)h(formal)h(series)f(with)h(r)-5 b(esp)g(e)g(ct)34 b(to)f(the)g(variable)g Fm(z)t Fd(,)1486 4491 y Fn(^)1460 4514 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))27 b(=)1872 4428 y Ff(X)1854 4624 y Fl(n)1897 4633 y Fk(1)1931 4624 y Fi(\025)p Fj(0)2072 4430 y Fn(^)2046 4453 y Fm(X)2121 4467 y Fl(n)2164 4476 y Fk(1)2203 4453 y Fn(\()p Fm(t)p Fn(\))p 2046 4493 261 4 v 2117 4577 a Fm(n)2172 4591 y Fj(1)2211 4577 y Fn(!)2317 4514 y Fm(z)2363 4477 y Fl(n)2406 4486 y Fk(1)2445 4514 y Fm(:)75 4830 y Fd(Then,)377 4807 y Fn(^)351 4830 y Fm(X)8 b Fn(\()p Fm(t;)15 b(z)t Fn(\))42 b Fd(is)g(a)g(formal)h(series)f(solution)h(of)f Fn(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\))42 b Fd(if)f(and)i(only)f(if)g (the)g(fol)5 b(lowing)42 b(di\013er)-5 b(ential)75 4943 y(r)g(elation)34 b Fn(\()p Fm(S)5 b(D)s(R)q Fn(\))33 b Fd(is)g(satis\014e)-5 b(d)194 5153 y Fn(^)168 5176 y Fm(X)243 5190 y 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526 y Fl(k)1818 535 y Fk(0)1853 526 y Fl(;k)1910 535 y Fk(1)1948 511 y Fn(\))1983 530 y Fj(\()p Fl(k)2047 539 y Fk(0)2082 530 y Fl(;k)2139 539 y Fk(1)2173 530 y Fj(\))p Fi(2S)2325 511 y Fh(2)f(P)7 b(g)p Fm(;)556 704 y Fh(M)665 718 y Fj(2)730 704 y Fn(=)25 b Fh(f)p Fn(\()p Fm(T)959 718 y Fj(1)999 704 y Fm(;)15 b Fn(\()p Fm(m)1154 719 y Fl(k)1191 728 y Fk(0)1226 719 y Fl(;k)1283 728 y Fk(1)1321 704 y Fn(\))1356 723 y Fj(\()p Fl(k)1420 732 y Fk(0)1456 723 y Fl(;k)1513 732 y Fk(1)1547 723 y Fj(\))p Fi(2S)1673 704 y Fn(\))25 b Fh(2)g Fg(N)1885 667 y Fj(1+Card\()p Fi(S)5 b Fj(\))2260 704 y Fn(:)2401 618 y Ff(X)2310 819 y Fj(\()p Fl(k)2374 828 y Fk(0)2409 819 y Fl(;k)2466 828 y Fk(1)2500 819 y Fj(\))p Fi(2S)2637 704 y Fm(m)2717 719 y Fl(k)2754 728 y Fk(0)2789 719 y Fl(;k)2846 728 y Fk(1)2904 704 y Fn(+)20 b Fm(T)3048 718 y Fj(1)3113 704 y Fn(=)25 b Fm(m)3289 718 y Fj(1)3328 704 y Fh(g)75 970 y Fd(and)672 1083 y Fh(M)781 1097 y Fj(3)846 1083 y Fn(=)g Fh(f)p Fn(\()p Fm(n)1077 1097 y Fj(1)1117 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Fd(We)32 b(assume)h(that)h(the)f(\014nite)g(set)g Fh(S)39 b Fd(satis\014es)1314 1776 y Fh(S)32 b(\032)25 b(f)p Fn(\()p Fm(k)1624 1790 y Fj(0)1665 1776 y Fm(;)15 b(k)1752 1790 y Fj(1)1792 1776 y Fn(\))26 b Fh(2)e Fg(N)2004 1738 y Fj(2)2069 1776 y Fn(:)h Fm(k)2166 1790 y Fj(1)2231 1776 y Fh(\024)g Fm(S)g Fh(\000)20 b Fn(1)p Fh(g)p Fm(:)75 1980 y Fd(Mor)-5 b(e)g(over,)35 b(assume)g(that)g(formal)h(series)1586 1957 y Fn(^)1560 1980 y Fm(X)1635 1994 y Fl(\026)1682 1980 y Fn(\()p Fm(t)p Fn(\))p Fd(,)f(for)f Fn(0)29 b Fh(\024)e Fm(\026)h Fh(\024)f Fm(S)f Fh(\000)21 b Fn(1)p Fd(,)35 b(ar)-5 b(e)35 b(given.)45 b(Then,)35 b(ther)-5 b(e)35 b(exists)f(a)75 2093 y(unique)i(se)-5 b(quenc)g(e)37 b Fn(\()807 2070 y(^)781 2093 y Fm(X)856 2107 y Fl(\026)903 2093 y Fn(\()p Fm(t)p Fn(\)\))1041 2107 y Fl(\026)p Fi(\025)p Fj(0)1215 2093 y Fd(of)h(formal)g(series)f(which)h(satis\014es)g(the)f (di\013er)-5 b(ential)38 b(r)-5 b(e)g(curr)g(enc)g(e)38 b Fn(\()p Fm(S)5 b(D)s(R)q Fn(\))p Fd(.)75 2206 y(As)38 b(a)h(r)-5 b(esult,)40 b(for)f(given)f(formal)i(series)e Fn(\()1597 2183 y(^)1571 2206 y Fm(X)1646 2220 y Fl(\026)1693 2206 y Fn(\()p Fm(t)p Fn(\)\))p Fd(,)j(for)e Fn(0)d Fh(\024)f Fm(\026)g Fh(\024)h Fm(S)29 b Fh(\000)24 b Fn(1)p Fd(,)40 b(ther)-5 b(e)39 b(exists)g(a)f(unique)g(formal)75 2319 y(series)33 b Fn(\()393 2296 y(^)367 2319 y Fm(X)8 b Fn(\))p Fd(,)32 b(solution)i(of)f Fn(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\))p Fd(.)75 2605 y Fo(4)135 b(Summabilit)l(y)57 b(of)f(formal)g(solutions)h(of)f(non-linear)g(partial)h(di\013er-)277 2754 y(en)l(tial)46 b(equations)g(with)g(deviating)g(v)-7 b(ariables)75 2957 y Fn(In)38 b(this)g(section,)k(w)m(e)d(describ)s(e)f (su\016cien)m(t)h(conditions)g(under)e(whic)m(h)h(the)g(formal)h (series)g(solutions)g(con-)75 3070 y(structed)30 b(in)g(the)h(previous) f(section)h(are)g(1)p Fh(\000)p Fn(summable.)216 3183 y(Firstly)-8 b(,)35 b(w)m(e)e(recall)i(estimates)f(for)f(high)f(order)h (deriv)-5 b(ativ)m(es)34 b(of)f(comp)s(ositions)h(of)f(functions)f (obtained)75 3296 y(as)k(a)g(consequence)g(of)g(the)g(higher)f(order)g (c)m(hain)h(rule)g(in)m(tro)s(duced)f(b)m(y)g(T.)h(Y)-8 b(amanak)j(a)37 b(in)e([20)q(],)j(Theorem)75 3409 y(2.1.)75 3622 y Fe(Lemma)d(2)45 b Fd(L)-5 b(et)36 b Fm(D)j Fd(b)-5 b(e)35 b(an)h(op)-5 b(en)37 b(disc)f(c)-5 b(enter)g(e)g(d)37 b(at)f(the)g(origin)g(in)g Fg(C)f Fd(and)h Fm(G)g Fd(b)-5 b(e)36 b(an)g(op)-5 b(en)37 b(set)e(in)h Fg(C)p Fd(.)50 b(L)-5 b(et)75 3734 y Fm(v)40 b Fn(:)c Fm(D)j Fh(!)e Fm(G)i Fd(and)h Fm(w)f Fn(:)d Fm(G)h Fh(!)f Fg(C)j Fd(b)-5 b(e)38 b(holomorphic)k(functions,)f(such)e(that)h Fh(j)p Fm(@)2809 3748 y Fl(x)2853 3734 y Fm(v)s Fn(\()p Fm(x)p Fn(\))p Fh(j)e(\024)e Fn(1)j Fd(for)g(al)5 b(l)40 b Fm(x)c Fh(2)g Fm(D)s Fd(.)75 3847 y(Then,)31 b(ther)-5 b(e)31 b(exist)g(an)g(op)-5 b(en)31 b(disc)g Fm(D)1370 3861 y Fj(1)1435 3847 y Fh(\032)25 b Fm(D)33 b Fd(c)-5 b(enter)g(e)g(d)31 b(at)g(the)g(origin)g(and)g Fm(r)d(>)d Fn(0)p Fd(,)31 b(such)f(that)i(the)e(estimates)75 3960 y(hold,)1057 4111 y Fh(j)p Fm(@)1135 4073 y Fl(n)1130 4133 y(x)1183 4111 y Fn(\()p Fm(w)23 b Fh(\016)e Fm(v)s Fn(\)\()p Fm(x)p Fn(\))p Fh(j)27 b(\024)1806 4025 y Ff(X)1724 4216 y Fl(a)1761 4225 y Fk(1)1796 4216 y Fj(+)p Fl(a)1888 4225 y Fk(2)1923 4216 y Fj(=)p Fl(n)2062 4049 y Fm(n)p Fn(!)p 2045 4090 113 4 v 2045 4173 a Fm(a)2093 4187 y Fj(1)2133 4173 y Fn(!)2168 4111 y Fh(j)p Fn(\()p Fm(@)2281 4073 y Fl(a)2318 4082 y Fk(1)2276 4133 y Fl(x)2358 4111 y Fm(w)r Fn(\)\()p Fm(v)s Fn(\()p Fm(x)p Fn(\)\))p Fh(j)p Fm(r)2768 4073 y Fl(a)2805 4082 y Fk(2)2847 4111 y Fm(;)75 4360 y Fd(for)33 b(al)5 b(l)33 b Fm(x)26 b Fh(2)e Fm(D)588 4374 y Fj(1)661 4360 y Fd(and)33 b Fm(n)25 b Fh(2)g Fg(N)p Fd(.)75 4572 y Fn(w)m(e)31 b(note)g(that)g(similar)g(computations)g(ha)m(v)m(e)g(b)s (een)f(made)g(in)g(the)h(pap)s(er)e([15)q(],)i(pp.)40 b(667{668.)216 4735 y(The)g(next)h(result)g(giv)m(es)h(su\016cien)m(t)f (conditions)g(for)f(the)h(existence)h(of)f(1)p Fh(\000)p Fn(summable)f(formal)h(series)75 4848 y(solutions)34 b(\()519 4825 y(^)493 4848 y Fm(X)8 b Fn(\))33 b(of)h(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\).)50 b(In)33 b(particular,)i(w)m(e)f(ha) m(v)m(e)g(to)g(giv)m(e)h(additional)g("shrinking")e(conditions)h(on)75 4961 y(the)d(functions)e Fm(\036)677 4976 y Fl(k)714 4985 y Fk(0)749 4976 y Fl(;k)806 4985 y Fk(1)874 4961 y Fn(to)i(get)h(summabilit)m(y)f(prop)s(erties)e(for)i(formal)f(series) h(solutions)g(of)f(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\).)75 5174 y Fe(Theorem)35 b(1)46 b Fd(Consider)33 b(the)g(semi-line)-5 b(ar)34 b(p)-5 b(artial)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quation)33 b(with)h(shrinkings,)75 5378 y Fn(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\))487 b Fm(@)967 5340 y Fl(S)962 5400 y(z)1044 5355 y Fn(^)1018 5378 y Fm(X)8 b Fn(\()p Fm(t;)15 b(z)t Fn(\))26 b(=)f Fm(A)p Fn(\()p Fm(t;)15 b(z)t(;)g Fn(\(\()p Fm(@)1797 5337 y Fl(k)1834 5346 y Fk(0)1792 5402 y Fl(t)1875 5378 y Fm(@)1928 5340 y Fl(k)1965 5349 y Fk(1)1923 5400 y Fl(z)2030 5355 y Fn(^)2004 5378 y Fm(X)7 b Fn(\)\()p Fm(\036)2210 5393 y Fl(k)2247 5402 y Fk(0)2283 5393 y Fl(;k)2340 5402 y Fk(1)2378 5378 y Fn(\()p Fm(t)p Fn(\))p Fm(;)15 b(z)t Fn(\)\))2637 5396 y Fj(\()p Fl(k)2701 5405 y Fk(0)2737 5396 y Fl(;k)2794 5405 y Fk(1)2828 5396 y Fj(\))p Fi(2S)2955 5378 y Fn(\))p Fm(;)75 5582 y Fd(wher)-5 b(e)941 5695 y Fh(S)32 b Fn(=)25 b Fh(f)p Fn(\()p Fm(k)1251 5709 y Fj(0)1292 5695 y Fm(;)15 b(k)1379 5709 y Fj(1)1419 5695 y Fn(\))26 b Fh(2)f Fg(N)1632 5657 y Fj(2)1696 5695 y Fn(:)g(2)p Fm(k)1838 5709 y Fj(0)1899 5695 y Fn(+)20 b Fm(k)2037 5709 y Fj(1)2102 5695 y Fh(\024)25 b Fm(S;)15 b Fn(\()p Fm(k)2376 5709 y Fj(0)2416 5695 y Fm(;)g(k)2503 5709 y Fj(1)2543 5695 y Fn(\))26 b Fh(6)p Fn(=)f(\(0)p Fm(;)15 b(S)5 b Fn(\))p Fh(g)p Fm(:)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 3809 100 a Fn(6)173 399 y Fd(a\))46 b(Assume)33 b(that)i(the)f(formal)h(series)1555 376 y Fn(^)1528 399 y Fm(X)1603 413 y Fl(\026)1650 399 y Fn(\()p Fm(t)p Fn(\))p Fd(,)f(for)g Fn(0)27 b Fh(\024)g Fm(\026)f Fh(\024)h Fm(S)e Fh(\000)c Fn(1)p Fd(,)34 b(ar)-5 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Fd(,)g(with)h Fm(\036)1083 1787 y Fl(k)1120 1796 y Fk(0)1154 1787 y Fl(;k)1211 1796 y Fk(1)1249 1772 y Fn(\(0\))e(=)d(0)p Fd(.)61 b(Mor)-5 b(e)g(over,)41 b(assume)e(that)h Fh(j)p Fm(@)2666 1786 y Fl(t)2696 1772 y Fm(\036)2750 1787 y Fl(k)2787 1796 y Fk(0)2821 1787 y Fl(;k)2878 1796 y Fk(1)2917 1772 y Fn(\()p Fm(t)p Fn(\))p Fh(j)c(\024)g Fn(1)j Fd(on)g Fm(O)j Fd(and)d(that)302 1884 y Fm(\036)356 1899 y Fl(k)393 1908 y Fk(0)428 1899 y Fl(;k)485 1908 y Fk(1)523 1884 y Fn(\()574 1861 y(\026)558 1884 y Fm(S)5 b Fn(\))26 b Fh(\032)791 1861 y Fn(\026)776 1884 y Fm(S)5 b Fd(,)32 b(for)h(al)5 b(l)33 b(close)-5 b(d)34 b(se)-5 b(ctor)1716 1861 y Fn(\026)1701 1884 y Fm(S)30 b Fh(\032)25 b Fm(G)33 b Fd(with)h(bise)-5 b(cting)32 b(dir)-5 b(e)g(ction)34 b Fm(d)p Fd(.)216 2060 y(Then,)c(the)f(formal)i(series)1190 2037 y Fn(^)1164 2060 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))p Fd(,)31 b(solution)f(of)f Fn(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\))p Fd(,)31 b(is)e Fn(1)p Fh(\000)p Fd(summable)h(with) f(r)-5 b(esp)g(e)g(ct)31 b(to)f Fm(t)e Fd(in)h(the)75 2173 y(dir)-5 b(e)g(ction)34 b Fm(d)26 b Fh(2)f Fg(R)p Fd(.)75 2401 y Fe(Pro)s(of)37 b Fn(The)f(goal)i(of)e(the)h(pro)s(of)e (is)i(to)g(sho)m(w)f(that)h(the)f(conditions)h Fe(iii\))f Fn(in)g(Prop)s(osition)g(1)h(are)g(ful\014lled.)75 2514 y(First)c(of)h(all,)h(from)d(the)i(pap)s(er)e([1],)j(Theorem)d(6,)j(w)m (e)e(deduce)g(that)h(the)f(comp)s(osition)h(of)f(a)h(1)p Fh(\000)p Fn(summable)75 2627 y(formal)27 b(series)623 2604 y(^)597 2627 y Fm(X)7 b Fn(\()p Fm(t)p Fn(\))28 b(in)e(direction)h Fm(d)p Fn(,)h(with)f(an)f(holomorphic)h(function)g Fm(\036)f Fn(whic)m(h)h(satis\014es)g(the)g(assumption)75 2740 y Fe(c)p Fn(\))q(,)240 2717 y(^)214 2740 y Fm(X)7 b Fn(\()p Fm(\036)p Fn(\()p Fm(t)p Fn(\)\))33 b(is)f(again)h(1)p Fh(\000)p Fn(summable)e(in)g(direction)i Fm(d)p Fn(.)44 b(F)-8 b(rom)33 b(general)f(classical)i(results,)e(see)g([4)q(],)g(w)m (e)h(also)75 2853 y(kno)m(w)26 b(that)h(deriv)-5 b(ation,)29 b(addition)d(and)g(pro)s(duct)f(also)j(conserv)m(e)f(1)p Fh(\000)p Fn(summabilit)m(y)g(of)g(formal)f(series.)40 b(F)-8 b(rom)75 2966 y(these)38 b(results,)h(the)e(assumption)g Fe(a)p Fn(\))g(on)g(initial)i(conditions,)h(the)d(relation)h(\()p Fm(S)5 b(D)s(R)q Fn(\))38 b(and)f(Prop)s(osition)g(2,)75 3079 y(w)m(e)d(deduce)g(that)g(the)g(formal)g(series)1441 3056 y(^)1415 3079 y Fm(X)1490 3093 y Fl(\026)1537 3079 y Fn(\()p Fm(t)p Fn(\))g(are)g(1)p Fh(\000)p Fn(summable)f(in)h(the)f (direction)i Fm(d)c Fh(2)f Fg(R)p Fn(,)35 b(for)e(all)i Fm(\026)30 b Fh(\025)h Fn(0.)75 3192 y(Moreo)m(v)m(er,)i(their)d(sums)f Fm(X)1020 3206 y Fl(\026)1067 3192 y Fn(\()p Fm(t)p Fn(\))i(are)g (holomorphic)f(in)g(the)h(sectorial)h(region)f Fm(G)p Fn(.)216 3305 y(The)j(rest)g(of)g(the)g(pro)s(of)g(will)g(consist)h(in) f(pro)m(ving)g(that)h(the)f(estimates)i(in)d Fe(iii\))h Fn(of)g(Prop)s(osition)h(1)f(are)75 3418 y(satis\014ed)c(for)h(the)f (formal)h(series)1276 3395 y(^)1249 3418 y Fm(X)1324 3432 y Fl(\026)1371 3418 y Fn(\()p Fm(t)p Fn(\).)42 b(W)-8 b(e)31 b(consider)f(no)m(w)h(the)f(sequence)h(of)g(p)s(ositiv)m(e)g (real)g(n)m(um)m(b)s(ers)1451 3596 y Fm(w)1516 3610 y Fl(n)1559 3619 y Fk(0)1594 3610 y Fl(;m)1676 3619 y Fk(1)1740 3596 y Fn(=)25 b(sup)1845 3684 y Fl(t)p Fi(2)1929 3667 y Fj(\026)1917 3684 y Fl(S)1988 3596 y Fh(j)p Fm(@)2066 3556 y Fl(n)2109 3565 y Fk(0)2061 3620 y Fl(t)2148 3596 y Fm(X)2223 3610 y Fl(m)2285 3619 y Fk(1)2324 3596 y Fn(\()p Fm(t)p Fn(\))p Fh(j)p Fm(;)75 3846 y Fn(for)37 b Fm(n)276 3860 y Fj(0)315 3846 y Fm(;)15 b(m)435 3860 y Fj(1)510 3846 y Fh(\025)36 b Fn(0,)j(where)1011 3823 y(\026)996 3846 y Fm(S)j Fn(is)37 b(a)g(closed)h(sector)f(in)g Fm(G)g Fn(with)g(bisecting)h(direction)f Fm(d)p Fn(.)61 b(F)-8 b(rom)37 b(the)g(Leibniz)75 3959 y(form)m(ula,)31 b(w)m(e)g(ha)m(v)m(e)g(that)158 4260 y Fm(A)226 4275 y Fl(k)263 4284 y Fk(0)298 4275 y Fl(;k)355 4284 y Fk(1)393 4260 y Fn(\()p Fm(t)p Fn(\))25 b(=)g Fm(@)670 4202 y Fl(\027)705 4214 y Fc(k)738 4229 y Fk(0)773 4214 y Fc(;k)825 4229 y Fk(1)665 4284 y Fl(t)868 4260 y Fn(\()903 4174 y Ff(X)909 4370 y Fi(M)993 4379 y Fk(3)1059 4197 y Fn(\()p Fm(@)1147 4156 y Fl(k)1184 4165 y Fk(0)1142 4221 y Fl(t)1224 4197 y Fm(X)1299 4212 y Fl(n)1342 4221 y Fk(1)1376 4212 y Fj(+)p Fl(k)1468 4221 y Fk(1)1507 4197 y Fn(\)\()p Fm(\036)1631 4212 y Fl(k)1668 4221 y Fk(0)1703 4212 y Fl(;k)1760 4221 y Fk(1)1798 4197 y Fn(\()p Fm(t)p Fn(\)\))p 1059 4239 878 4 v 1438 4322 a Fm(n)1493 4336 y Fj(1)1533 4322 y Fn(!)1962 4260 y Fh(\001)15 b(\001)g(\001)2093 4162 y Fn(\()p Fm(@)2181 4122 y Fl(k)2218 4131 y Fk(0)2176 4186 y Fl(t)2257 4162 y Fm(X)2332 4181 y Fj(\()p Fl(n)2402 4192 y Fc(T)2438 4209 y(k)2471 4224 y Fk(0)2507 4209 y Fc(;k)2559 4224 y Fk(1)2602 4181 y Fj(+)p Fl(k)2694 4190 y Fk(1)2728 4181 y Fj(\))2759 4162 y Fn(\)\()p Fm(\036)2883 4177 y Fl(k)2920 4186 y Fk(0)2956 4177 y Fl(;k)3013 4186 y Fk(1)3051 4162 y Fn(\()p Fm(t)p Fn(\)\))p 2093 4239 1097 4 v 2464 4322 a(\()p Fm(n)2554 4336 y Fl(T)2595 4348 y Fc(k)2628 4363 y Fk(0)2663 4348 y Fc(;k)2715 4363 y Fk(1)2758 4322 y Fn(\)!)3200 4260 y(\))177 4623 y(=)273 4537 y Ff(X)279 4733 y Fi(M)363 4742 y Fk(3)420 4537 y Ff(X)426 4733 y Fi(M)510 4742 y Fk(4)728 4561 y Fm(\027)773 4576 y Fl(k)810 4585 y Fk(0)844 4576 y Fl(k)881 4585 y Fk(1)919 4561 y Fn(!)p 576 4602 521 4 v 576 4685 a Fm(\026)631 4699 y Fj(1)670 4685 y Fn(!)g Fh(\001)g(\001)g(\001)i Fm(\026)887 4699 y Fl(T)928 4711 y Fc(k)961 4726 y Fk(0)995 4711 y Fc(k)1028 4726 y Fk(1)1071 4685 y Fn(!)1116 4560 y Fm(@)1169 4515 y Fl(\026)1211 4524 y Fk(1)1164 4584 y Fl(t)1251 4560 y Fn([\()p Fm(@)1364 4519 y Fl(k)1401 4528 y Fk(0)1359 4584 y Fl(t)1440 4560 y Fm(X)1515 4575 y Fl(n)1558 4584 y Fk(1)1593 4575 y Fj(+)p Fl(k)1685 4584 y Fk(1)1723 4560 y Fn(\)\()p Fm(\036)1847 4575 y Fl(k)1884 4584 y Fk(0)1919 4575 y Fl(;k)1976 4584 y Fk(1)2015 4560 y Fn(\()p Fm(t)p Fn(\)\)])p 1116 4602 1063 4 v 1588 4685 a Fm(n)1643 4699 y Fj(1)1682 4685 y Fn(!)2204 4623 y Fh(\001)e(\001)g(\001)2335 4525 y Fm(@)2388 4451 y Fl(\026)2430 4462 y Fc(T)2466 4479 y(k)2499 4494 y Fk(0)2535 4479 y Fc(k)2568 4494 y Fk(1)2383 4549 y Fl(t)2615 4525 y Fn([\()p Fm(@)2728 4485 y Fl(k)2765 4494 y Fk(0)2723 4549 y Fl(t)2804 4525 y Fm(X)2879 4544 y Fj(\()p Fl(n)2949 4555 y Fc(T)2985 4572 y(k)3018 4587 y Fk(0)3054 4572 y Fc(;k)3106 4587 y Fk(1)3148 4544 y Fj(+)p Fl(k)3240 4553 y Fk(1)3275 4544 y Fj(\))3306 4525 y Fn(\)\()p Fm(\036)3430 4540 y Fl(k)3467 4549 y Fk(0)3502 4540 y Fl(;k)3559 4549 y Fk(1)3597 4525 y Fn(\()p Fm(t)p Fn(\)\)])p 2335 4602 1427 4 v 2871 4685 a(\()p Fm(n)2961 4699 y Fl(T)3002 4711 y Fc(k)3035 4726 y Fk(0)3070 4711 y Fc(;k)3122 4726 y Fk(1)3165 4685 y Fn(\)!)75 4889 y(where)746 5002 y Fh(M)855 5016 y Fj(3)920 5002 y Fn(=)25 b Fh(f)p Fn(\()p Fm(n)1151 5016 y Fj(1)1191 5002 y Fm(;)15 b(:)g(:)g(:)h(;)f(n)1447 5016 y Fl(T)1488 5028 y Fc(k)1521 5043 y Fk(0)1556 5028 y Fc(k)1589 5043 y Fk(1)1632 5002 y Fn(\))25 b Fh(2)g Fg(N)1844 4964 y Fl(T)1885 4976 y Fc(k)1918 4991 y Fk(0)1952 4976 y Fc(k)1985 4991 y Fk(1)2053 5002 y Fn(:)h Fm(n)2159 5016 y Fj(1)2218 5002 y Fn(+)20 b Fh(\001)15 b(\001)g(\001)21 b Fn(+)f Fm(n)2581 5016 y Fl(T)2622 5028 y Fc(k)2655 5043 y Fk(0)2689 5028 y Fc(k)2722 5043 y Fk(1)2790 5002 y Fn(=)25 b Fm(m)2966 5017 y Fl(k)3003 5026 y Fk(0)3037 5017 y Fl(k)3074 5026 y Fk(1)3113 5002 y Fh(g)p Fm(;)763 5175 y Fh(M)872 5189 y Fj(4)937 5175 y Fn(=)g Fh(f)p Fn(\()p Fm(\026)1168 5189 y Fj(1)1208 5175 y Fm(;)15 b(:)g(:)g(:)h(;)f(\026)1464 5189 y Fl(T)1505 5201 y Fc(k)1538 5216 y Fk(0)1573 5201 y Fc(k)1606 5216 y Fk(1)1649 5175 y Fn(\))26 b Fh(2)e Fg(N)1861 5138 y Fl(T)1902 5150 y Fc(k)1935 5165 y Fk(0)1970 5150 y Fc(k)2003 5165 y Fk(1)2071 5175 y Fn(:)h Fm(\026)2176 5189 y Fj(1)2236 5175 y Fn(+)20 b Fh(\001)15 b(\001)g(\001)21 b Fn(+)f Fm(\026)2599 5189 y Fl(T)2640 5201 y Fc(k)2673 5216 y Fk(0)2707 5201 y Fc(k)2740 5216 y Fk(1)2808 5175 y Fn(=)25 b Fm(\027)2949 5190 y Fl(k)2986 5199 y Fk(0)3020 5190 y Fl(k)3057 5199 y Fk(1)3096 5175 y Fh(g)p Fm(:)75 5329 y Fn(By)k(the)f(lemma)h(2)f(and) g(the)g(assumption)g Fe(c)p Fn(\))q(,)h(w)m(e)f(deduce)g(that)h(for)f (all)h Fm(k)2613 5343 y Fj(0)2653 5329 y Fm(;)15 b(k)2740 5343 y Fj(1)2780 5329 y Fn(,)29 b(there)f(exist)i Fm(r)3319 5344 y Fl(k)3356 5353 y Fk(0)3390 5344 y Fl(;k)3447 5353 y Fk(1)3510 5329 y Fm(>)25 b Fn(0)k(suc)m(h)75 5442 y(that,)417 5659 y Fh(j)p Fm(@)495 5614 y Fl(\026)537 5622 y Fc(m)490 5683 y Fl(t)601 5659 y Fn([\()p Fm(@)714 5618 y Fl(k)751 5627 y Fk(0)709 5683 y Fl(t)790 5659 y Fm(X)865 5674 y Fl(n)908 5682 y Fc(m)967 5674 y Fj(+)p Fl(k)1059 5683 y Fk(1)1097 5659 y Fn(\)\()p Fm(\036)1221 5674 y Fl(k)1258 5683 y Fk(0)1294 5674 y Fl(;k)1351 5683 y Fk(1)1389 5659 y Fn(\()p Fm(t)p Fn(\)\)])p Fh(j)d(\024)1811 5572 y Ff(X)1699 5764 y Fl(a)1736 5773 y Fk(1)1771 5764 y Fj(+)p Fl(a)1863 5773 y Fk(2)1898 5764 y Fj(=)p Fl(\026)1995 5772 y Fc(m)2080 5597 y Fm(\026)2135 5611 y Fl(m)2201 5597 y Fn(!)p 2080 5638 147 4 v 2096 5721 a Fm(a)2144 5735 y Fj(1)2184 5721 y Fn(!)2236 5659 y Fh(j)p Fn(\()p Fm(@)2349 5618 y Fl(a)2386 5627 y Fk(1)2422 5618 y Fj(+)p Fl(k)2514 5627 y Fk(0)2344 5683 y Fl(t)2552 5659 y Fm(X)2627 5674 y Fl(n)2670 5682 y Fc(m)2729 5674 y Fj(+)p Fl(k)2821 5683 y Fk(1)2860 5659 y Fn(\()p Fm(\036)2949 5674 y Fl(k)2986 5683 y Fk(0)3020 5674 y Fl(;k)3077 5683 y Fk(1)3116 5659 y Fn(\()p Fm(t)p Fn(\)\))p Fh(j)p Fm(r)3323 5618 y Fl(a)3360 5627 y Fk(2)3320 5688 y Fl(k)3357 5697 y Fk(0)3392 5688 y Fl(;k)3449 5697 y Fk(1)3487 5659 y Fm(:)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3809 100 a Fn(7)75 399 y(F)-8 b(rom)31 b(the)f(latter)i(inequalities,)g(w)m(e)f(deduce)489 703 y Fh(j)p Fm(A)582 718 y Fl(k)619 727 y Fk(0)653 718 y Fl(;k)710 727 y Fk(1)748 703 y Fn(\()p Fm(t)p Fn(\))p Fh(j)c(\024)998 616 y Ff(X)1005 813 y Fi(M)1089 822 y Fk(3)1145 616 y Ff(X)1151 813 y Fi(M)1235 822 y Fk(4)1291 703 y Fm(\027)1336 718 y Fl(k)1373 727 y Fk(0)1408 718 y Fl(k)1445 727 y Fk(1)1483 703 y Fn(!)1523 570 y Fl(T)1564 582 y Fc(k)1597 597 y Fk(0)1632 582 y Fc(k)1665 597 y Fk(1)1556 616 y Ff(Y)1537 810 y Fl(m)p Fj(=1)1704 703 y Fn(\()1851 616 y Ff(X)1739 808 y Fl(a)1776 817 y Fk(1)1811 808 y Fj(+)p Fl(a)1903 817 y Fk(2)1938 808 y Fj(=)p Fl(\026)2035 816 y Fc(m)2119 639 y Fh(j)p Fn(\()p Fm(@)2232 599 y Fl(a)2269 608 y Fk(1)2305 599 y Fj(+)p Fl(k)2397 608 y Fk(0)2227 663 y Fl(t)2436 639 y Fm(X)2511 654 y Fl(n)2554 662 y Fc(m)2612 654 y Fj(+)p Fl(k)2704 663 y Fk(1)2743 639 y Fn(\()p Fm(\036)2832 654 y Fl(k)2869 663 y Fk(0)2904 654 y Fl(;k)2961 663 y Fk(1)2999 639 y Fn(\()p Fm(t)p Fn(\)\))p Fh(j)p 2119 682 1044 4 v 2512 765 a Fm(a)2560 779 y Fj(1)2599 765 y Fn(!)p Fm(n)2679 779 y Fl(m)2745 765 y Fn(!)3173 703 y Fm(r)3217 662 y Fl(a)3254 671 y Fk(2)3214 732 y Fl(k)3251 741 y Fk(0)3285 732 y Fl(;k)3342 741 y Fk(1)3380 703 y Fn(\))p Fm(:)75 994 y Fn(If)j(w)m(e)h(put)574 1296 y Fm(B)643 1311 y Fl(k)680 1320 y Fk(0)714 1311 y Fl(;k)771 1320 y Fk(1)809 1296 y Fn(\()p Fm(t)p Fn(\))26 b(=)1034 1209 y Ff(X)1040 1406 y Fi(M)1124 1415 y Fk(3)1190 1232 y Fn(\()p Fm(@)1278 1192 y Fl(k)1315 1201 y Fk(0)1273 1256 y Fl(t)1355 1232 y Fm(X)1430 1247 y Fl(n)1473 1256 y Fk(1)1507 1247 y Fj(+)p Fl(k)1599 1256 y Fk(1)1638 1232 y Fn(\)\()p Fm(\036)1762 1247 y Fl(k)1799 1256 y Fk(0)1834 1247 y Fl(;k)1891 1256 y Fk(1)1929 1232 y Fn(\()p Fm(t)p Fn(\)\))p 1190 1275 878 4 v 1569 1358 a Fm(n)1624 1372 y Fj(1)1663 1358 y Fn(!)2093 1296 y Fh(\001)15 b(\001)g(\001)2224 1198 y Fn(\()p Fm(@)2312 1158 y Fl(k)2349 1167 y Fk(0)2307 1222 y Fl(t)2388 1198 y Fm(X)2463 1217 y Fj(\()p Fl(n)2533 1228 y Fc(T)2569 1245 y(k)2602 1260 y Fk(0)2638 1245 y Fc(;k)2690 1260 y Fk(1)2732 1217 y Fj(+)p Fl(k)2824 1226 y Fk(1)2859 1217 y Fj(\))2890 1198 y Fn(\)\()p Fm(\036)3014 1213 y Fl(k)3051 1222 y Fk(0)3087 1213 y Fl(;k)3144 1222 y Fk(1)3182 1198 y Fn(\()p Fm(t)p Fn(\)\))p 2224 1275 1097 4 v 2595 1358 a(\()p Fm(n)2685 1372 y Fl(T)2726 1384 y Fc(k)2759 1399 y Fk(0)2794 1384 y Fc(;k)2846 1399 y Fk(1)2889 1358 y Fn(\)!)3330 1296 y Fm(;)75 1587 y Fn(w)m(e)31 b(get,)h(again)f(from)f(the)g(Leibniz)h(form)m(ula,)158 1816 y Fm(@)211 1776 y Fl(n)254 1785 y Fk(0)206 1840 y Fl(t)293 1816 y Fn(\()p Fm(b)367 1835 y Fl(T)408 1844 y Fk(1)443 1835 y Fl(;)p Fj(\()p Fl(T)531 1847 y Fc(k)564 1862 y Fk(0)599 1847 y Fc(k)632 1862 y Fk(1)670 1835 y Fj(\))697 1849 y Fk(\()p Fc(k)754 1864 y Fk(0)789 1849 y Fc(;k)841 1864 y Fk(1)876 1849 y(\))p Fb(2S)991 1816 y Fn(\()p Fm(t)p Fn(\)\005)1162 1835 y Fj(\()p Fl(k)1226 1844 y Fk(0)1261 1835 y Fl(;k)1318 1844 y Fk(1)1352 1835 y Fj(\))p Fi(2S)1479 1816 y Fm(B)1548 1831 y Fl(k)1585 1840 y Fk(0)1619 1831 y Fl(;k)1676 1840 y Fk(1)1714 1816 y Fn(\()p Fm(t)p Fn(\)\))1113 2022 y(=)1208 1936 y Ff(X)1215 2133 y Fi(M)1299 2142 y Fk(5)1696 1961 y Fm(n)1751 1975 y Fj(0)1790 1961 y Fn(!)p 1365 2001 782 4 v 1365 2084 a Fm(\027)1410 2098 y Fj(1)1449 2084 y Fn(!\005)1542 2103 y Fj(\()p Fl(k)1606 2112 y Fk(0)1641 2103 y Fl(;k)1698 2112 y Fk(1)1732 2103 y Fj(\))p Fi(2S)1859 2084 y Fn(\()p Fm(\027)1939 2099 y Fl(k)1976 2108 y Fk(0)2010 2099 y Fl(k)2047 2108 y Fk(1)2086 2084 y Fn(!\))2156 2022 y Fm(@)2209 1982 y Fl(\027)2244 1991 y Fk(1)2204 2046 y Fl(t)2283 2022 y Fm(b)2322 2041 y Fl(T)2363 2050 y Fk(1)2398 2041 y Fl(;)p Fj(\()p Fl(T)2486 2053 y Fc(k)2519 2068 y Fk(0)2553 2053 y Fc(k)2586 2068 y Fk(1)2625 2041 y Fj(\))2652 2055 y Fk(\()p Fc(k)2709 2070 y Fk(0)2744 2055 y Fc(;k)2796 2070 y Fk(1)2830 2055 y(\))p Fb(2S)2946 2022 y Fn(\()p Fm(t)p Fn(\)\005)3117 2041 y Fj(\()p Fl(k)3181 2050 y Fk(0)3216 2041 y Fl(;k)3273 2050 y Fk(1)3307 2041 y Fj(\))p Fi(2S)3433 2022 y Fm(A)3501 2037 y Fl(k)3538 2046 y Fk(0)3573 2037 y Fl(;k)3630 2046 y Fk(1)3668 2022 y Fn(\()p Fm(t)p Fn(\))75 2314 y(where)619 2427 y Fh(M)728 2441 y Fj(5)793 2427 y Fn(=)25 b Fh(f)p Fn(\()p Fm(\027)1014 2441 y Fj(1)1054 2427 y Fm(;)15 b Fn(\()p Fm(\027)1174 2442 y Fl(k)1211 2451 y Fk(0)1246 2442 y Fl(k)1283 2451 y Fk(1)1321 2427 y Fn(\))1356 2445 y Fj(\()p Fl(k)1420 2454 y Fk(0)1456 2445 y Fl(;k)1513 2454 y Fk(1)1547 2445 y Fj(\))p Fi(2S)1673 2427 y Fn(\))25 b Fh(2)g Fg(N)1885 2389 y Fj(1+Card\()p Fi(S)5 b Fj(\))2260 2427 y Fn(:)25 b Fm(\027)2355 2441 y Fj(1)2415 2427 y Fn(+)2596 2340 y Ff(X)2506 2542 y Fj(\()p Fl(k)2570 2551 y Fk(0)2605 2542 y Fl(;k)2662 2551 y Fk(1)2696 2542 y Fj(\))p Fi(2S)2833 2427 y Fm(\027)2878 2442 y Fl(k)2915 2451 y Fk(0)2949 2442 y Fl(k)2986 2451 y Fk(1)3050 2427 y Fn(=)g Fm(n)3201 2441 y Fj(0)3240 2427 y Fh(g)p Fm(:)75 2693 y Fn(On)30 b(the)g(other)h(hand,)e(b)m(y)h(classical)j(estimates,)f(w)m(e)f(ha)m (v)m(e)h(that)1452 2878 y Fm(n)1507 2892 y Fj(0)1546 2878 y Fn(!)p 1259 2919 507 4 v 1259 3002 a Fm(a)1307 3016 y Fj(1)1346 3002 y Fn(!)15 b Fh(\001)g(\001)g(\001)i Fm(a)1556 3016 y Fl(T)1597 3028 y Fc(k)1630 3043 y Fk(0)1664 3028 y Fc(k)1697 3043 y Fk(1)1740 3002 y Fn(!)1801 2940 y Fh(\024)2158 2878 y Fn(\(2)p Fm(n)2293 2892 y Fj(0)2333 2878 y Fn(\)!)p 1907 2919 740 4 v 1907 3002 a(\(2)p Fm(a)2035 3016 y Fj(1)2075 3002 y Fn(\)!)e Fh(\001)g(\001)g(\001)i Fn(\(2)p Fm(a)2400 3016 y Fl(T)2441 3028 y 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Fk(1)850 3342 y Fl(;)p Fj(\()p Fl(T)938 3354 y Fc(k)971 3369 y Fk(0)1005 3354 y Fc(;k)1057 3369 y Fk(1)1096 3342 y Fj(\))1123 3356 y Fk(\()p Fc(k)1180 3371 y Fk(0)1215 3356 y Fc(;k)1267 3371 y Fk(1)1301 3356 y(\))p Fb(2S)1413 3342 y Fl(;\027)1468 3351 y Fk(1)1506 3324 y Fn(\005)1574 3342 y Fj(\()p Fl(k)1638 3351 y Fk(0)1673 3342 y Fl(;k)1730 3351 y Fk(1)1763 3342 y Fj(\))p Fi(2S)1905 3141 y Ff(0)1905 3305 y(@)1984 3238 y(X)1991 3434 y Fi(M)2075 3443 y Fk(3)2131 3238 y Ff(X)2137 3434 y Fi(M)2221 3443 y Fk(4)2277 3324 y Fn(\005)2345 3266 y Fl(T)2386 3278 y Fc(k)2419 3293 y Fk(0)2454 3278 y Fc(k)2487 3293 y Fk(1)2345 3350 y Fl(m)p Fj(=1)2530 3324 y Fn(\()2677 3238 y Ff(X)2565 3429 y Fl(a)2602 3438 y Fk(1)2637 3429 y Fj(+)p Fl(a)2729 3438 y Fk(2)2764 3429 y Fj(=)p Fl(\026)2861 3437 y Fc(m)2945 3261 y Fm(v)2989 3276 y Fl(a)3026 3285 y Fk(1)3062 3276 y Fj(+)p Fl(k)3154 3285 y Fk(0)3188 3276 y Fl(;n)3251 3284 y Fc(m)3309 3276 y Fj(+)p Fl(k)3401 3285 y Fk(1)p 2945 3303 494 4 v 3005 3386 a Fn(\(2)p Fm(a)3133 3400 y Fj(1)3173 3386 y Fn(\)!)p Fm(n)3288 3400 y Fl(m)3355 3386 y Fn(!)3449 3324 y Fm(r)3493 3283 y Fl(a)3530 3292 y Fk(2)3490 3353 y Fl(k)3527 3362 y Fk(0)3562 3353 y Fl(;k)3619 3362 y Fk(1)3657 3324 y Fn(\))3692 3141 y Ff(1)3692 3305 y(A)75 3612 y Fn(for)30 b(all)h Fm(n)395 3626 y Fj(0)434 3612 y Fm(;)15 b(m)554 3626 y Fj(1)619 3612 y Fh(\025)25 b Fn(0,)31 b(where)1378 3725 y Fh(M)1487 3687 y Fi(0)1487 3747 y Fj(1)1552 3725 y Fn(=)25 b Fh(f)p Fn(\()p Fm(T)1781 3740 y Fl(k)1818 3749 y Fk(0)1853 3740 y Fl(;k)1910 3749 y Fk(1)1948 3725 y Fn(\))1983 3743 y Fj(\()p Fl(k)2047 3752 y Fk(0)2082 3743 y Fl(;k)2139 3752 y Fk(1)2173 3743 y Fj(\))p Fi(2S)2325 3725 y Fh(2)f(P)7 b(g)p Fm(;)544 3907 y Fh(M)653 3921 y Fj(2)718 3907 y Fn(=)24 b Fh(f)p Fn(\()p Fm(T)946 3921 y Fj(1)987 3907 y Fm(;)15 b Fn(\()p Fm(m)1142 3922 y Fl(k)1179 3931 y Fk(0)1214 3922 y Fl(;k)1271 3931 y Fk(1)1309 3907 y Fn(\))1344 3926 y Fj(\()p Fl(k)1408 3935 y Fk(0)1443 3926 y Fl(;k)1500 3935 y Fk(1)1534 3926 y Fj(\))p Fi(2S)1660 3907 y Fn(\))26 b Fh(2)f Fg(N)1873 3870 y Fj(1+Card)o(\()p Fi(S)5 b Fj(\))2247 3907 y Fn(:)2388 3821 y Ff(X)2298 4022 y Fj(\()p Fl(k)2362 4031 y Fk(0)2397 4022 y Fl(;k)2454 4031 y Fk(1)2488 4022 y Fj(\))p Fi(2S)2625 3907 y Fm(m)2705 3922 y Fl(k)2742 3931 y Fk(0)2776 3922 y Fl(;k)2833 3931 y Fk(1)2891 3907 y Fn(+)20 b Fm(T)3035 3921 y Fj(1)3100 3907 y Fn(=)25 b Fm(m)3276 3921 y Fj(1)3315 3907 y Fh(g)p Fm(;)619 4196 y Fh(M)728 4210 y Fj(5)793 4196 y Fn(=)g Fh(f)p Fn(\()p Fm(\027)1014 4210 y Fj(1)1054 4196 y Fm(;)15 b Fn(\()p Fm(\027)1174 4211 y Fl(k)1211 4220 y Fk(0)1246 4211 y Fl(k)1283 4220 y Fk(1)1321 4196 y Fn(\))1356 4214 y Fj(\()p Fl(k)1420 4223 y Fk(0)1456 4214 y Fl(;k)1513 4223 y Fk(1)1547 4214 y Fj(\))p Fi(2S)1673 4196 y Fn(\))25 b Fh(2)g Fg(N)1885 4158 y Fj(1+Card\()p Fi(S)5 b Fj(\))2260 4196 y Fn(:)25 b Fm(\027)2355 4210 y Fj(1)2415 4196 y Fn(+)2596 4109 y Ff(X)2506 4311 y Fj(\()p Fl(k)2570 4320 y Fk(0)2605 4311 y Fl(;k)2662 4320 y Fk(1)2696 4311 y Fj(\))p Fi(2S)2833 4196 y Fm(\027)2878 4211 y Fl(k)2915 4220 y Fk(0)2949 4211 y Fl(k)2986 4220 y Fk(1)3050 4196 y Fn(=)g Fm(n)3201 4210 y Fj(0)3240 4196 y Fh(g)p Fm(;)672 4471 y Fh(M)781 4485 y Fj(3)846 4471 y Fn(=)g Fh(f)p Fn(\()p Fm(n)1077 4485 y Fj(1)1117 4471 y Fm(;)15 b(:)g(:)g(:)h(;)f(n)1373 4490 y Fj(\()p Fl(T)1441 4502 y Fc(k)1474 4517 y Fk(0)1509 4502 y Fc(;k)1561 4517 y Fk(1)1600 4490 y Fj(\))1632 4471 y Fn(\))25 b Fh(2)g Fg(N)1844 4434 y Fl(T)1885 4446 y Fc(k)1918 4461 y Fk(0)1952 4446 y Fc(k)1985 4461 y Fk(1)2053 4471 y Fn(:)h Fm(n)2159 4485 y Fj(1)2218 4471 y Fn(+)20 b Fh(\001)15 b(\001)g(\001)21 b Fn(+)f Fm(n)2581 4490 y Fj(\()p Fl(T)2649 4502 y Fc(k)2682 4517 y Fk(0)2717 4502 y Fc(;k)2769 4517 y Fk(1)2807 4490 y Fj(\))2864 4471 y Fn(=)25 b Fm(m)3040 4486 y Fl(k)3077 4495 y Fk(0)3111 4486 y Fl(k)3148 4495 y Fk(1)3187 4471 y Fh(g)p Fm(;)763 4651 y Fh(M)872 4665 y Fj(4)937 4651 y Fn(=)g Fh(f)p Fn(\()p Fm(\026)1168 4665 y Fj(1)1208 4651 y Fm(;)15 b(:)g(:)g(:)h(;)f(\026)1464 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b(c)m(ho)s(ose)h(no)m(w)e(the)h(latter)g(sequences)g(as)g (follo)m(ws.)1626 5740 y Fm(v)1670 5754 y Fl(n)1713 5763 y Fk(0)1748 5754 y Fl(;m)1830 5763 y Fk(1)1894 5740 y Fn(:=)25 b Fm(w)2080 5754 y Fl(n)2123 5763 y Fk(0)2157 5754 y Fl(;m)2239 5763 y Fk(1)2278 5740 y Fm(;)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fn(9)75 399 y(for)30 b(all)h(0)26 b Fh(\024)f Fm(m)587 413 y Fj(1)652 399 y Fh(\024)g Fm(S)g Fh(\000)20 b Fn(1,)31 b Fm(n)1076 413 y Fj(0)1140 399 y Fh(\025)25 b Fn(0.)216 511 y(F)-8 b(rom)31 b(the)g(assumption)e Fe(b)p Fn(\))i(of)g(Theorem)f(1,)h(w)m(e) g(deduce)f(that)h(the)f(series)1505 698 y Fm(E)5 b Fn(\()p Fm(t;)15 b Fn(0\))p Fm(;)g(:)g(:)g(:)j(;)d(@)2021 660 y Fl(S)t Fi(\000)p Fj(1)2016 720 y Fl(z)2163 698 y Fm(E)5 b Fn(\()p Fm(t;)15 b Fn(0\))75 884 y(are)41 b(con)m(v)m(ergen)m(t)j(in) c(a)h(neigh)m(b)s(orho)s(o)s(d)f(of)h(the)g(origin.)73 b(Moreo)m(v)m(er,)46 b(due)40 b(to)i(the)f(relations)h(\()p Fm(M)10 b(R)q(I)d Fn(\))41 b(and)75 997 y(\()p Fm(M)10 b(R)q 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y Fn(!)p Fm(;)75 1764 y Fn(for)30 b(all)h Fm(n)395 1778 y Fj(0)434 1764 y Fm(;)15 b(m)554 1778 y Fj(1)619 1764 y Fh(\025)25 b Fn(0.)216 1877 y Fd(Final)5 b(ly,)32 b(we)f(c)-5 b(onclude)31 b(that)h(the)g(c)-5 b(ondition)32 b Fe(iii\))e Fd(in)h(Pr)-5 b(op)g(osition)33 b(1)f(is)e(satis\014e)-5 b(d)33 b(for)e(the)g(formal)h(series)101 1967 y Fn(^)75 1990 y Fm(X)150 2004 y Fl(n)193 2013 y Fk(1)232 1990 y Fn(\()p Fm(t)p Fn(\),)e(so)g(that)g(the)f(formal)h(series)1406 1967 y(^)1379 1990 y Fm(X)8 b Fn(\()p Fm(t;)15 b(z)t Fn(\),)31 b(solution)f(of)f(\()p Fm(S)5 b(P)13 b(D)s(E)5 b Fn(\),)31 b(is)e(1)p Fh(\000)p Fn(summable)g(with)g(resp)s(ect)h(to)g Fm(t)75 2103 y Fn(in)g(the)h(direction)g Fm(d)25 b Fh(2)g Fg(R)p Fn(.)2790 b Fa(2)75 2431 y Fo(5)135 b(Application)60 b(to)h(non-linear)g(adv)-7 b(anced-argumen)l(t)60 b(partial)i (di\013er-)277 2580 y(en)l(tial)46 b(equations)75 2783 y Fn(Consider)30 b(the)g(follo)m(wing)i(Mo)s(ebius)e(transformation,) 1667 3006 y Fm(\036)p Fn(\()p Fm(t)p Fn(\))25 b(=)2039 2945 y Fm(Qt)p 1955 2985 273 4 v 1955 3068 a Fn(1)c(+)f Fm(H)7 b(t)2238 3006 y(;)75 3226 y Fn(de\014ned)27 b(on)g Fg(C)15 b Fh(n)g(f\000)p Fn(1)p Fm(=H)7 b Fh(g)p Fn(,)30 b(with)e(0)d Fm(<)g(Q)h(<)f Fn(1)j(and)f Fm(H)32 b Fh(2)25 b Fg(R)2111 3240 y Fj(+)2170 3226 y Fn(.)39 b(It)28 b(is)g(easy)h(to)f (see)g(that)h(the)f(image)h Fm(\036)p Fn(\()p Fm(D)s Fn(\))f(of)g(a)75 3339 y(line)e Fm(D)j Fn(whic)m(h)d(go)s(es)h(through) e(the)i(p)s(oin)m(t)f Fm(t)f Fn(=)g(0)h(and)g(is)g(not)g(equal)h(to)g Fg(R)e Fn(in)h Fg(C)p Fn(,)h(is)f(a)h(circle)g(whic)m(h)f(is)g(tangen)m (t)75 3452 y(to)34 b Fm(D)h Fn(and)e(go)s(es)g(through)f(the)i(p)s(oin) m(ts)e Fm(t)e Fn(=)f(0)k(and)g Fm(t)c Fn(=)g Fm(Q=H)7 b Fn(.)49 b(Moreo)m(v)m(er,)36 b(there)d(exists)h(a)f(neigh)m(b)s(orho) s(o)s(d)75 3565 y Fm(O)g Fn(of)e(the)f(origin)h(in)f Fg(C)g Fn(suc)m(h)g(that)1439 3751 y Fh(j)p Fm(\036)p Fn(\()p Fm(t)p Fn(\))p Fh(j)c(\024)f(j)p Fm(t)p Fh(j)61 b Fm(;)76 b Fh(j)p Fm(@)2086 3765 y Fl(t)2116 3751 y Fm(\036)p 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b(August)m(yno)m(wicz,)50 b Fd(On)c(some)g(gener)-5 b(alize)g(d)48 b(V)-7 b(olterr)i(a)47 b(c)-5 b(ondition)48 b(and)f(advanc)-5 b(e)g(d)48 b(ar)-5 b(gument)47 b(for)267 2060 y(inte)-5 b(gr)g(al)34 b(e)-5 b(quations)p Fn(,)32 b(Commen)m(t.)f(Math.)g(Prace)g(Mat.)g Fe(40)g Fn(\(2000\),)i(7{17.)75 2244 y([3])97 b(A.)29 b(August)m(yno)m(wicz,)i(H.)e(Leszczynski,)h(W.)f(W)-8 b(alter,)32 b Fd(Cauchy-Kovalevskaya)h(the)-5 b(ory)32 b(for)g(non-line)-5 b(ar)267 2357 y(e)g(quations)32 b(with)g(deviating) g(variables)p Fn(,)f(Nonlinear)e(Anal.)h Fe(45)f Fn(\(2001\),)j(no.)d (6,)h(Ser.)f(A:)g(Theory)g(Meth-)267 2470 y(o)s(ds,)h(743-753.)75 2654 y([4])97 b(W.)39 b(Balser,)j Fd(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 2767 y(e)g(quations)p Fn(,)32 b(Springer-V)-8 b(erlag,)31 b(New-Y)-8 b(ork,)32 b(2000.)75 2952 y([5])97 b(W.)38 b(Balser,)j(S.)c(Malek,)42 b Fd(F)-7 b(ormal)41 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