Content-Type: multipart/mixed; boundary="-------------0702270703186" This is a multi-part message in MIME format. ---------------0702270703186 Content-Type: text/plain; name="07-47.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-47.keywords" Cauchy problem, partial differential equation, Gevrey estimates, distribution, hyperfunction, analytic continuation ---------------0702270703186 Content-Type: application/postscript; name="stairstokes.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="stairstokes.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: stairstokes.dvi %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 CMMI10 MSBM10 CMBX12 CMR8 %%+ CMMI8 CMSY8 CMEX10 CMR6 CMSY10 CMMI6 CMTI10 LASY10 CMSY6 MSBM7 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o stairstokes.ps stairstokes.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2007.02.27:1149 %%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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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 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 65 /A put dup 66 /B put dup 67 /C put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S 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 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: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % 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 (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 /minus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. 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: 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 14 /ffi put dup 18 /grave put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright 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 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 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright 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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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 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 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 92 /intersectiondisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 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 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 91 /union 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: 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 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma 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 65 /A put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 76 /L put dup 80 /P put dup 82 /R 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 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 109 /m put dup 110 /n put dup 111 /o put dup 112 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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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 33 /exclam 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 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 (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 18 /theta put dup 20 /kappa put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 31 /chi put dup 32 /psi 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 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N 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 88 /X put dup 97 /a 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 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 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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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. 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 1 /Delta put dup 3 /Lambda 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 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 47 /slash 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 59 /semicolon 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 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 dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 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 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 98 /b 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 dup 121 /y put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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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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68 83 57 71 47 33 68 71 59 62 69 66 64 68 3[71 1[25 25 45 45 45 45 45 45 45 45 45 45 45 25 30 25 71 1[35 35 25 5[25 10[45 2[45 45 3[76 1[51 53 7[63 1[76 57{}86 90.9091 /CMR10 rf /Fu 188[60 67[{}1 83.022 /MSBM10 rf /Fv 133[39 5[30 56[23 35[41 23[{}4 83.022 /CMMI10 rf /Fw 133[37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 1[23 46 42 25 37 46 37 46 42 9[85 3[46 5[76 52 14[65 12[42 1[23 28 23 29[69 1[46 48 11[{}37 83.022 /CMR10 rf /Fx 139[37 38 39 14[42 53 46 31[72 65[{}7 83.022 /CMBX10 rf /Fy 134[51 51 1[51 54 38 38 38 51 54 1[54 81 27 51 1[27 54 1[30 43 54 43 54 49 10[73 73 70 54 72 4[89 61 5[64 2[70 1[73 76 6[49 1[49 49 49 49 49 49 49 49 1[27 33 27 4[27 19[49 19[{}46 99.6264 /CMR12 rf /Fz 134[70 1[96 1[73 51 52 51 70 73 66 73 111 36 70 1[36 73 66 40 58 73 58 1[66 13[73 3[103 33[43 33[75 11[{}25 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 351 763 a Fz(On)44 b(the)g(Stok)l(es)f 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4626 y Fq(\()p Fp(k)r Fq(\))1584 4659 y Fk(2)g(D)1756 4626 y Fn(0)1779 4659 y Ft(\()p Fi(R)1880 4673 y Fq(+)1939 4659 y Ft(\))p Fh(,)k(wher)-5 b(e)41 b Fr(F)51 b Fk(2)39 b Fr(L)2579 4626 y Fq(1)2618 4659 y Ft(\()p Fi(R)2719 4673 y Fq(+)2778 4659 y Ft(\))h Fh(and)h Ft(supp)n(\()p Fr(F)13 b Ft(\))40 b Fk(\032)e Ft([)p Fr(k)s(;)15 b Ft(+)p Fk(1)p Ft(\))p Fh(.)75 4772 y(Then)42 b Fr(f)51 b Fh(is)42 b(a)g(stair)-5 b(c)g(ase)43 b(distribution)g(and)g(the)f(de)-5 b(c)g(omp)g(osition)45 b(of)d Fr(f)51 b Fh(has)43 b(the)f(fol)5 b(lowing)42 b(terms)h Ft(\001)3703 4786 y Fq(0)3784 4772 y Ft(=)75 4885 y(\001)151 4899 y Fq(1)229 4885 y Ft(=)38 b Fr(:)15 b(:)g(:)40 b Ft(=)e(\001)668 4900 y Fp(k)r Fn(\000)p Fq(1)839 4885 y Ft(=)g(0)p Fh(,)k Ft(\001)1139 4900 y Fp(k)1220 4885 y Ft(=)c Fr(F)13 b Ft(1)1445 4903 y Fq([)p Fp(k)r(;k)r Fq(+1])1716 4885 y Fh(and)41 b(for)f Fr(n)e Fk(\025)h Ft(1)p Fh(,)j Ft(\001)2446 4900 y Fp(k)r Fq(+)p Fp(n)2624 4885 y Ft(=)c Fr(G)2804 4899 y Fp(n)2852 4885 y Ft(1)2897 4903 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Ft(\001)1410 5598 y Fp(k)r Fq(+)p Fp(n)1551 5583 y Fk(jj)1601 5597 y Fp(\027)1670 5583 y Fk(\024)g Fr(e)1808 5545 y Fq(\(2)p Fp(\027)t Fn(\000)p Fp(n\027)t Fq(\))2146 5521 y Fr(n)2201 5488 y Fp(n)p Fn(\000)p Fq(1)p 2088 5562 308 4 v 2088 5645 a Ft(\()p Fr(n)20 b Fk(\000)g Ft(1\)!)2405 5583 y Fk(jj)p Fr(F)13 b Fk(jj)2576 5597 y Fp(\027)2620 5583 y Fr(:)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Ft(4)75 399 y Fo(Prop)s(osition)36 b(2)46 b Fh(T)-7 b(ake)42 b Fr(\027)48 b(>)42 b(\027)1219 413 y Fq(0)1300 399 y Fr(>)g Ft(1)g Fh(and)h(let)f Fr(f)52 b Fk(2)41 b Fr(L)2084 413 y Fp(\027)2119 422 y Fl(0)2158 399 y Ft(\()p Fi(R)2259 413 y Fq(+)2318 399 y Ft(\))p Fh(.)70 b(Then)42 b Fr(f)51 b Fh(b)-5 b(elongs)43 b(to)f Fk(D)3305 366 y Fn(0)3302 421 y Fp(\027)3388 399 y Fh(and)g(the)h(de-) 75 511 y(c)-5 b(omp)g(osition)38 b(\(3\))f(of)e Fr(f)45 b Fh(has)36 b(the)g(fol)5 b(lowing)37 b(terms)f Ft(\001)1974 525 y Fp(n)2051 511 y Ft(=)30 b Fr(G)2223 525 y Fp(n)2271 511 y Ft(1)2316 530 y Fq([)p Fp(n;n)p Fq(+1])2590 511 y Fh(with)37 b Fr(G)2862 525 y Fp(n)2939 511 y Ft(=)31 b Fk(P)7 b Ft(\()p Fr(G)3217 525 y Fp(n)p Fn(\000)p Fq(1)3355 511 y Ft(1)3400 530 y Fq([)p Fp(n;)p Fq(+)p Fn(1)p Fq(\))3640 511 y Ft(\))35 b Fh(and)75 624 y Fr(G)146 638 y Fq(0)215 624 y Ft(=)30 b Fr(f)10 b Fh(,)34 b(for)i Fr(n)29 b Fk(\025)g Ft(0)p Fh(.)49 b(Mor)-5 b(e)g(over,)37 b(ther)-5 b(e)35 b(exists)h(a)f(c)-5 b(onstant)37 b Fr(C)k Fh(which)36 b(dep)-5 b(ends)37 b(only)e(on)h Fr(\027)3327 638 y Fq(0)3366 624 y Fr(;)15 b(\027)41 b Fh(such)35 b(that)75 737 y Fk(jj)p Fr(f)10 b Fk(jj)230 752 y Fp(\027;d)351 737 y Fk(\024)25 b Fr(C)7 b Fk(jj)p Fr(f)j Fk(jj)674 751 y Fp(\027)709 760 y Fl(0)747 737 y Fh(.)75 950 y Fo(Prop)s(osition)36 b(3)46 b Fh(L)-5 b(et)29 b Fi(N)e Fh(b)-5 b(e)29 b(the)f(set)h(of)f(p) -5 b(ositive)30 b(inte)-5 b(gers)28 b Fr(n)d Fk(\025)g Ft(1)p Fh(.)41 b(The)28 b(set)h Fk(D)s Ft(\()p Fi(R)2918 964 y Fq(+)2987 950 y Fk(n)11 b Fi(N)p Ft(\))29 b Fh(of)f Fr(C)3347 917 y Fn(1)3421 950 y Fk(\000)p Fh(functions)75 1063 y(with)33 b(c)-5 b(omp)g(act)35 b(supp)-5 b(ort)35 b(in)e Fi(R)1123 1077 y Fq(+)1201 1063 y Fk(n)21 b Fi(N)32 b Fh(is)h(dense)g(in)f Fk(D)1898 1030 y Fn(0)1895 1085 y Fp(\027)1939 1063 y Fh(.)75 1275 y Fo(Prop)s(osition)k(4)46 b Fh(F)-7 b(or)30 b(al)5 b(l)30 b Fr(f)5 b(;)1140 1251 y Ft(~)1121 1275 y Fr(f)34 b Fk(2)25 b(D)1359 1242 y Fn(0)1356 1298 y Fp(\027)1399 1275 y Fh(,)30 b(we)f(have)h Fr(f)22 b Fk(\003)1939 1251 y Ft(~)1919 1275 y Fr(f)35 b Fk(2)24 b(D)2157 1242 y Fn(0)2154 1298 y Fp(\027)2198 1275 y Fh(.)40 b(Mor)-5 b(e)g(over,)31 b(ther)-5 b(e)30 b(exists)g Fr(C)i(>)25 b Ft(0)k Fh(\(dep)-5 b(ending)75 1388 y(on)33 b Fr(\027)6 b Fh(\))32 b(such)h(that)1410 1501 y Fk(jj)p Fr(f)d Fk(\003)1620 1477 y Ft(~)1601 1501 y Fr(f)9 b Fk(jj)1705 1516 y Fp(\027;d)1826 1501 y Fk(\024)25 b Fr(C)7 b Fk(jj)p Fr(f)j Fk(jj)2149 1516 y Fp(\027;d)2244 1501 y Fk(jj)2314 1477 y Ft(~)2294 1501 y Fr(f)g Fk(jj)2399 1516 y Fp(\027;d)2494 1501 y Fr(;)75 1668 y Fh(for)33 b(al)5 b(l)33 b Fr(f)5 b(;)460 1644 y Ft(~)440 1668 y Fr(f)34 b Fk(2)25 b(D)678 1635 y Fn(0)675 1690 y Fp(\027)719 1668 y Fh(.)75 1880 y Ft(In)30 b(the)g(pap)s(er,)g(w)m(e)h(will)f (sometimes)i(denote)f(b)m(y)f Fr(@)1844 1847 y Fn(\000)p Fq(1)1839 1903 y Fp(r)1939 1880 y Fr(f)10 b Ft(\()p Fr(r)s Ft(\))29 b(the)i(con)m(v)m(olution)h Fk(H)21 b(\003)g Fr(f)39 b Ft(for)31 b(all)g Fr(f)j Fk(2)25 b(D)3533 1847 y Fn(0)3530 1903 y Fp(\027)3574 1880 y Ft(.)75 2093 y Fo(Prop)s(osition)36 b(5)46 b Fh(L)-5 b(et)33 b Fr(\027)e(>)25 b Ft(1)33 b Fh(b)-5 b(e)32 b(lar)-5 b(ge)34 b(enough)f(such)f(that)i Fr(\027)6 b(e)2270 2060 y Fq(1)p Fn(\000)p Fp(\027)2429 2093 y Fr(<)25 b Ft(1)33 b Fh(and)g(let)g Fr(h)g Fh(b)-5 b(e)32 b(a)h Fr(C)3254 2060 y Fn(1)3329 2093 y Fk(\000)p Fh(function)f(on)75 2206 y Fi(R)141 2220 y Fq(+)232 2206 y Fh(such)h(that)h(ther)-5 b(e)33 b(exist)g(c)-5 b(onstants)35 b Fr(C)1528 2221 y Fp(h)1598 2206 y Fr(>)25 b Ft(0)p Fr(;)15 b(\026)26 b(>)f Ft(0)33 b Fh(and)g Fr(\032)25 b(>)g 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Fn(\000)p Fq(1)1250 5658 y Ft(\()p Fr(r)s Ft(\))p Fr(@)1417 5620 y Fp(k)r Fn(\000)p Fq(2)1412 5680 y Fp(r)1550 5658 y Fr( )s Ft(\()p Fr(r)s Ft(\)])1751 5620 y Fp(k)r Fq(+1)1751 5687 y Fp(k)1905 5658 y Ft(+)g Fr(:)15 b(:)g(:)21 b Ft(+)f(\()p Fk(\000)p Ft(1\)[)p Fr(G)2495 5672 y Fq(1)2536 5658 y Ft(\()p Fr(r)s Ft(\))p Fr( )s Ft(\()p Fr(r)s Ft(\)])2851 5620 y Fp(k)r Fq(+1)2851 5687 y Fp(k)3006 5658 y Ft(+)3096 5534 y Fm(Z)3187 5560 y Fp(k)r Fq(+1)3147 5740 y Fp(k)3335 5658 y Fr(f)10 b Ft(\()p Fr(r)s Ft(\))p Fr( )s Ft(\()p Fr(r)s Ft(\))p Fr(dr)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 3764 100 a Ft(11)75 399 y(Moreo)m(v)m(er,)33 b(as)d(a)h(telescopic)i(sum,)c(w)m(e)i(notice)h(that)75 667 y(\(26\))943 553 y Fn(1)913 581 y Fm(X)914 778 y Fp(k)r Fq(=1)1044 667 y Ft(\()p Fk(\000)p Ft(1\))1230 630 y Fp(k)1274 667 y Ft([)p Fr(G)1370 682 y Fp(k)1413 667 y Ft(\()p Fr(r)s Ft(\))p Fr(@)1580 630 y Fp(k)r Fn(\000)p Fq(1)1575 690 y Fp(r)1714 667 y Fr( )s Ft(\()p Fr(r)s Ft(\)])1915 629 y Fp(k)r Fq(+1)1915 696 y Fp(k)2074 667 y Ft(=)2170 581 y Fm(X)2171 778 y Fp(k)r Fq(=1)2316 667 y Fr(G)2387 682 y Fp(k)2430 667 y Ft(\()p Fr(k)s Ft(\)\()p Fr(@)2638 630 y Fp(k)r Fn(\000)p Fq(1)2633 690 y Fp(r)2773 667 y Fr( )s Ft(\)\()p Fr(k)s Ft(\))p Fr(:)75 955 y Ft(But,)c(b)m(y)f (construction,)i(w)m(e)e(ha)m(v)m(e)i(that)e Fr(G)1546 970 y Fp(k)1589 955 y Ft(\()p Fr(k)s Ft(\))g(=)e(0,)j(for)f(all)h Fr(k)g Fk(\025)d Ft(1,)k(so)e(that)h(this)f(latter)h(sum)e(\(26\))j(is) e(equal)75 1068 y(to)k(zero.)216 1181 y(In)f(conclusion,)h(from)f (\(25\))i(and)e(\(26\),)i(w)m(e)f(deduce)f(that)956 1353 y Fq(+)p Fn(1)953 1380 y Fm(X)954 1578 y Fp(k)r Fq(=0)1084 1466 y Ft(\()p Fk(\000)p Ft(1\))1270 1429 y Fp(k)1329 1343 y Fm(Z)1420 1369 y Fp(k)r Fq(+1)1379 1549 y Fp(k)1567 1466 y Ft(\001)1643 1481 y Fp(k)1686 1466 y Ft(\()p Fr(r)s Ft(\))p Fr(@)1853 1429 y Fp(k)1848 1489 y(r)1896 1466 y Fr( )s Ft(\()p Fr(r)s Ft(\))p Fr(dr)f Ft(=)2285 1343 y Fm(Z)2376 1369 y Fq(+)p Fn(1)2335 1549 y Fq(0)2521 1466 y Fr(f)10 b Ft(\()p Fr(r)s Ft(\))p Fr( )s Ft(\()p Fr(r)s Ft(\))p Fr(dr)m(:)3756 1727 y Fg(2)75 2063 y Fs(5)135 b(Cauc)l(h)l(y)45 b(problems)g(in)g(Banac)l(h)g(spaces)g(of)g (distributions)h Ff(D)3420 2020 y Fe(0)3417 2093 y Fv(\027)75 2266 y Ft(W)-8 b(e)32 b(consider)e(follo)m(wing)i(the)e(in)m (tegro-di\013eren)m(tial)k(equation)75 2551 y(\(27\))265 b Fr(@)553 2514 y Fp(S)548 2574 y(z)604 2551 y Fr(W)13 b Ft(\()p Fr(r)m(;)i(z)t Ft(\))27 b(=)1032 2438 y Fp(S)t Fn(\000)p Fq(1)1035 2465 y Fm(X)1020 2663 y Fp(k)1057 2672 y Fl(1)1091 2663 y Fq(=0)1196 2551 y Fr(h)1248 2566 y Fp(k)1285 2575 y Fl(1)1324 2551 y Ft(\()p Fr(r)s Ft(\))1453 2428 y Fm(Z)1544 2454 y Fp(r)1504 2634 y Fq(0)1598 2428 y Fm(Z)1688 2454 y Fp(r)1720 2463 y Fl(1)1648 2634 y Fq(0)1774 2551 y Fr(H)1850 2566 y Fp(k)1887 2575 y Fl(1)1925 2551 y Ft(\()p Fr(r)2001 2565 y Fq(1)2061 2551 y Fk(\000)20 b Fr(r)2193 2565 y Fq(2)2233 2551 y Fr(;)15 b(z)t Ft(\))p Fr(@)2407 2514 y Fp(k)2444 2523 y Fl(1)2402 2574 y Fp(z)2483 2551 y Fr(W)e Ft(\()p Fr(r)2658 2565 y Fq(2)2698 2551 y Fr(;)i(z)t Ft(\))p Fr(dr)2907 2565 y Fq(2)2948 2551 y Fr(dr)3036 2565 y Fq(1)3096 2551 y Ft(+)k Fr(d)p Ft(\()p Fr(r)m(;)c(z)t Ft(\))75 2844 y(with)30 b(initial)i(conditions)75 3048 y(\(28\))956 b(\()p Fr(@)1279 3010 y Fp(j)1274 3070 y(z)1317 3048 y Fr(W)13 b Ft(\)\()p Fr(r)m(;)i Ft(0\))27 b(=)e Fr(W)1853 3062 y Fp(j)1889 3048 y Ft(\()p Fr(r)s Ft(\))61 b Fr(;)76 b Ft(0)26 b Fk(\024)f Fr(j)30 b Fk(\024)25 b Fr(S)h Fk(\000)19 b Ft(1)p Fr(:)75 3252 y Ft(Let)31 b Fr(\027)g(>)25 b Ft(1)31 b(b)s(e)e(a)i(real)g(n)m(um)m(b)s(er)e(suc)m (h)h(that)h Fr(\027)6 b(e)1688 3219 y Fq(1)p Fn(\000)p Fp(\027)1847 3252 y Fr(<)25 b Ft(1)30 b(and)g Fr(S)g Fk(\025)25 b Ft(2)31 b(an)f(in)m(teger.)43 b Fh(We)32 b(assume)h(that)75 3415 y Fo(\(d1\))d Ft(F)-8 b(or)31 b(all)g(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1,)31 b Fr(W)1276 3429 y Fp(j)1312 3415 y Ft(\()p Fr(r)s Ft(\))26 b Fk(2)f(D)1611 3382 y Fn(0)1608 3437 y Fp(\027)1651 3415 y Ft(.)75 3528 y Fo(\(d2\))30 b Ft(F)-8 b(or)31 b(all)g(0)26 b Fk(\024)f Fr(k)800 3542 y Fq(1)865 3528 y Fk(\024)g Fr(S)g Fk(\000)20 b Ft(1,)957 3813 y Fr(H)1033 3828 y Fp(k)1070 3837 y Fl(1)1109 3813 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)1429 3699 y Fq(+)p Fn(1)1426 3727 y Fm(X)1429 3921 y Fp(q)r Fq(=0)1572 3813 y Fr(H)1648 3828 y Fp(k)1685 3837 y Fl(1)1719 3828 y Fp(;q)1777 3813 y Ft(\()p Fr(r)s Ft(\))p Fr(z)1937 3775 y Fp(q)1975 3813 y Fr(;)76 b(;)g(d)p Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))28 b(=)2545 3699 y Fq(+)p Fn(1)2542 3727 y Fm(X)2545 3921 y Fp(q)r Fq(=0)2688 3813 y Fr(d)2735 3827 y Fp(q)2773 3813 y Ft(\()p Fr(r)s Ft(\))p Fr(z)2933 3775 y Fp(q)75 4112 y Ft(are)j(holomorphic)f(functions)g(in)g Fr(z)35 b Ft(on)30 b(a)h(disc)f Fr(D)1781 4126 y Fp(R)1839 4112 y Ft(,)h Fr(R)26 b(>)f Ft(0,)31 b(with)f(v)-5 b(alues)30 b(in)g Fk(D)2843 4079 y Fn(0)2840 4135 y Fp(\027)2884 4112 y Ft(.)75 4225 y Fo(\(d3\))37 b Ft(There)g(exist)h Fr(\026;)15 b(\024)38 b(>)f Ft(1)p Fr(=\027)6 b Ft(,)40 b(suc)m(h)d(that)h(for)f(all)i(0)e Fk(\024)g Fr(k)2228 4239 y Fq(1)2305 4225 y Fk(\024)g Fr(S)30 b Fk(\000)24 b Ft(1,)40 b Fr(h)2756 4240 y Fp(k)2793 4249 y Fl(1)2832 4225 y Ft(\()p Fr(r)s Ft(\))e(are)g Fr(C)3215 4192 y Fn(1)3289 4225 y Fk(\000)p Ft(functions)f(on)75 4338 y(\(0)p Fr(;)15 b Ft(+)p Fk(1)p Ft(\))31 b(suc)m(h)g(that)1391 4494 y Fk(j)p Fr(h)1468 4446 y Fq(\()p Fp(q)r Fq(\))1468 4523 y Fp(k)1505 4532 y Fl(1)1561 4494 y Ft(\()p Fr(r)s Ft(\))p Fk(j)26 b(\024)f Fr(C)1887 4509 y Fp(k)1924 4518 y Fl(1)2203 4432 y Fr(q)s Ft(!)p 1972 4473 532 4 v 1972 4556 a(\()p Fr(\024)p Ft(\()p Fr(r)f Ft(+)c Fr(\026)p Ft(\)\))2375 4530 y Fq(1+)p Fp(q)2514 4494 y Fr(;)75 4709 y Ft(for)30 b(a)h(constan)m(t)h Fr(C)722 4724 y Fp(k)759 4733 y Fl(1)822 4709 y Fr(>)25 b Ft(0,)31 b(for)f(all)i Fr(q)c Fk(\025)d Ft(0)30 b(and)g(all)h Fr(r)d Fk(2)d Ft(\(0)p Fr(;)15 b Ft(+)p Fk(1)p Ft(\).)216 4822 y(The)29 b(follo)m(wing)h(lemma)g(is)f(a)g(consequence)h(of)f(the)g(h)m(yp)s (otheses)g Fo(\(d1\))p Ft(,)g Fo(\(d2\))p Ft(,)h Fo(\(d3\))e Ft(and)h(Prop)s(ositions)75 4935 y(4,)i(5.)75 5148 y Fo(Lemma)k(4)45 b Fh(Consider)34 b(a)f(formal)h(series)1495 5406 y Fr(W)13 b Ft(\()p Fr(r)m(;)i(z)t Ft(\))27 b(=)1911 5319 y Fm(X)1912 5517 y Fp(k)r Fn(\025)p Fq(0)2057 5406 y Fr(W)2143 5421 y Fp(k)2186 5406 y Ft(\()p Fr(r)s Ft(\))2310 5344 y Fr(z)2356 5311 y Fp(k)p 2310 5385 89 4 v 2317 5468 a Fr(k)s Ft(!)2409 5406 y Fr(;)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 3764 100 a Ft(12)75 399 y Fh(wher)-5 b(e)34 b Fr(W)418 414 y Fp(k)460 399 y Ft(\()p Fr(r)s Ft(\))26 b Fk(2)e(D)758 366 y Fn(0)755 421 y Fp(\027)799 399 y Fh(,)32 b(for)h(al)5 b(l)33 b Fr(k)28 b Fk(\025)d Ft(0)p Fh(.)42 b(Then,)33 b(the)g(series)f Fr(W)13 b Ft(\()p Fr(r)m(;)i(z)t Ft(\))34 b Fh(is)e(a)h(solution)h(of)e(\(27\),)i (for)f(initial)g(data)75 511 y Fr(W)161 525 y Fp(j)197 511 y Ft(\()p Fr(r)s Ft(\))26 b Fk(2)f(D)496 478 y Fn(0)493 534 y Fp(\027)536 511 y Fh(,)32 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fh(,)33 b(if)f(and)i(only)f(if)g(the)g(fol)5 b(lowing)33 b(r)-5 b(e)g(cursion)34 b(is)f(satis\014e)-5 b(d)75 795 y Ft(\(29\))669 733 y Fr(W)755 748 y Fp(k)r Fq(+)p Fp(S)899 733 y Ft(\()p Fr(r)s Ft(\))p 669 774 345 4 v 803 857 a Fr(k)s Ft(!)1048 795 y(=)1156 681 y Fp(S)t Fn(\000)p Fq(1)1159 709 y Fm(X)1144 906 y Fp(k)1181 915 y Fl(1)1215 906 y Fq(=0)1395 709 y Fm(X)1321 906 y Fp(q)1353 915 y Fl(1)1387 906 y Fq(+)p Fp(q)1474 915 y Fl(2)1508 906 y Fq(=)p Fp(k)1616 795 y Fr(h)1668 810 y Fp(k)1705 819 y Fl(1)1744 795 y Ft(\()p Fr(r)s Ft(\))p Fr(@)1911 757 y Fn(\000)p Fq(1)1906 817 y Fp(r)2006 795 y Ft(\()p Fr(H)2117 810 y Fp(k)2154 819 y Fl(1)2188 810 y Fp(;q)2240 819 y Fl(1)2278 795 y Ft(\()p Fr(r)s Ft(\))21 b Fk(\003)2488 732 y Fr(W)2574 747 y Fp(q)2606 756 y Fl(2)2640 747 y Fq(+)p Fp(k)2732 756 y Fl(1)2771 732 y Ft(\()p Fr(r)s Ft(\))p 2488 774 397 4 v 2634 857 a Fr(q)2675 871 y Fq(2)2714 857 y Ft(!)2895 795 y(\))f(+)g Fr(d)3088 810 y Fp(k)3131 795 y Ft(\()p Fr(r)s Ft(\))p Fr(;)75 1087 y Fh(for)38 b(al)5 b(l)38 b Fr(k)f Fk(\025)d Ft(0)p Fh(.)56 b(In)38 b(p)-5 b(articular,)40 b(for)e(given)f(initial)h(data)h Fr(W)2208 1101 y Fp(j)2244 1087 y Ft(\()p Fr(r)s Ft(\))c Fk(2)e(D)2560 1054 y Fn(0)2557 1109 y Fp(\027)2600 1087 y Fh(,)39 b Ft(0)34 b Fk(\024)g Fr(j)39 b Fk(\024)34 b Fr(S)29 b Fk(\000)23 b Ft(1)p Fh(,)39 b(ther)-5 b(e)38 b(exists)g(a)75 1200 y(unique)32 b(formal)i(series)f Fr(W)13 b Ft(\()p Fr(r)m(;)i(z)t Ft(\))34 b Fh(of)e(the)h(latter)h(form,)g(solution)f(of)g(\(27\))h(and) g(\(28\).)216 1410 y Ft(In)c(the)g(follo)m(wing,)i(w)m(e)f(giv)m(e)h (some)f(notations.)42 b(Let)31 b Fr(w)2109 1366 y Fp(\027;d)2107 1437 y(j)2230 1410 y Ft(=)25 b Fk(jj)p Fr(W)2462 1424 y Fp(j)2498 1410 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2662 1425 y Fp(\027;d)2758 1410 y Ft(,)31 b(0)26 b Fk(\024)f Fr(j)30 b Fk(\024)25 b Fr(S)h Fk(\000)20 b Ft(1)30 b(and)736 1635 y Fk(j)p Fr(H)7 b Fk(j)869 1590 y Fp(\027;d)869 1664 y(k)906 1673 y Fl(1)965 1635 y Ft(\()p Fr(z)t Ft(\))26 b(=)1203 1548 y Fm(X)1207 1744 y Fp(q)r Fn(\025)p Fq(0)1349 1635 y Fk(jj)p Fr(H)1475 1650 y Fp(k)1512 1659 y Fl(1)1547 1650 y Fp(;q)1604 1635 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)1768 1650 y Fp(\027;d)1864 1635 y Fr(z)1910 1597 y Fp(q)2009 1635 y Fr(;)76 b Fk(j)p Fr(d)p Fk(j)2207 1597 y Fp(\027;d)2303 1635 y Ft(\()p Fr(z)t Ft(\))27 b(=)2542 1548 y Fm(X)2545 1744 y Fp(q)r Fn(\025)p Fq(0)2688 1635 y Fk(jj)p Fr(d)2785 1649 y Fp(q)2824 1635 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2988 1650 y Fp(\027;d)3084 1635 y Fr(z)3130 1597 y Fp(q)3168 1635 y Fr(:)75 1954 y Fo(Prop)s(osition)36 b(9)46 b Fh(Ther)-5 b(e)44 b(exist)f(two)i(c)-5 b(onstants)45 b Fr(R)1893 1921 y Fn(0)1916 1954 y Fr(;)15 b(C)51 b(>)45 b Ft(0)e Fh(\(dep)-5 b(ending)45 b(on)f Fr(C)2953 1969 y Fp(k)2990 1978 y Fl(1)3028 1954 y Fh(,)i Fr(\024)p Fh(,)g Fr(\026)p Fh(,)g Fr(\027)6 b Fh(,)45 b Fk(j)p Fr(H)7 b Fk(j)3614 1909 y Fp(\027;d)3614 1983 y(k)3651 1992 y Fl(1)3710 1954 y Ft(\()p Fr(z)t Ft(\))p Fh(,)75 2067 y Ft(0)26 b Fk(\024)f Fr(k)289 2081 y Fq(1)354 2067 y Fk(\024)g Fr(S)g Fk(\000)20 b Ft(1)p Fh(\),)33 b(such)g(that)75 2180 y Fo(i\))i Fh(The)g(formal)i(series)e Fr(W)13 b Ft(\()p Fr(r)m(;)i(z)t Ft(\))p Fh(,)37 b(solution)f(of)g(the)f(pr)-5 b(oblem)37 b(\(27\))f(with)g(\(28\),)h(c)-5 b(onstructe)g(d)37 b(in)e(L)-5 b(emma)36 b(4,)75 2292 y(is)d(a)g(c)-5 b(onver)g(gent)33 b(series)g(in)f Fr(z)37 b Fh(on)c Fr(D)1353 2308 y Fp(R)1406 2289 y Fd(0)1434 2292 y Fh(,)f(with)h(c)-5 b(o)g(e\016cients)34 b(in)e Fk(D)2339 2259 y Fn(0)2336 2315 y Fp(\027)2380 2292 y Fh(.)75 2405 y Fo(ii\))g Fh(The)h(fol)5 b(lowing)34 b(estimates)f(hold)75 2685 y Ft(\(30\))1038 2571 y Fq(+)p Fn(1)1035 2599 y Fm(X)1036 2796 y Fp(k)r Fq(=0)1191 2622 y Fk(jj)p Fr(W)1327 2637 y Fp(k)1370 2622 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)1534 2637 y Fp(\027;d)p 1191 2664 439 4 v 1373 2747 a Fr(k)s Ft(!)1640 2685 y(\()p Fr(R)1745 2648 y Fn(0)1769 2685 y Ft(\))1804 2648 y Fp(k)1872 2685 y Fk(\024)25 b Fr(C)7 b Ft(\()2075 2571 y Fp(S)t Fn(\000)p Fq(1)2078 2599 y Fm(X)2082 2794 y Fp(j)t 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Fr(S)540 4714 y Fp(\034)t(;\016)653 4699 y Fk([)755 4676 y Ft(\026)734 4699 y Fr(D)809 4713 y Fq(1)848 4699 y Ft(.)41 b(In)30 b(the)g(follo)m(wing,)i(w)m(e)f(will)g(need)f(a)h(lemma)g(from)f([5],)h (Corollary)g(4.9.)75 4922 y Fo(Lemma)k(5)45 b Fh(L)-5 b(et)35 b Fo(F)p Ft(\()p Fr(t)p Ft(\))p Fr(;)15 b Fo(G)p Ft(\()p Fr(t)p Ft(\))31 b Fk(2)d(A)1289 4937 y Fp(\034)t(;\016)o(;\027) 1436 4922 y Fh(.)48 b(Then,)35 b(the)g(function)g Fr(t)29 b Fk(7!)2474 4849 y Fm(R)2534 4875 y Fp(t)2517 4954 y Fq(0)2579 4922 y Fo(F)p Ft(\()p Fr(t)22 b Fk(\000)f Fr(\034)10 b Ft(\))p Fo(G)p Ft(\()p Fr(\034)g Ft(\))p Fr(d\034)46 b Fh(b)-5 b(elong)35 b(to)h Fk(A)3708 4937 y Fp(\034)t(;\016)o(;\027)75 5035 y Fh(and)e(the)f(fol)5 b(lowing)33 b(ine)-5 b(quality)33 b(holds)951 5291 y Fk(jj)1016 5167 y Fm(Z)1107 5193 y Fp(t)1067 5373 y Fq(0)1152 5291 y Fo(F)p Ft(\()p Fr(t)20 b Fk(\000)g Fr(\034)10 b Ft(\))p Fo(G)p Ft(\()p Fr(\034)g Ft(\))p Fr(d\034)g Fk(jj)1831 5306 y Fp(\034)t(;\016)o(;\027)2006 5291 y Fk(\024)25 b(jj)p Fo(F)p Ft(\()p Fr(t)p Ft(\))p Fk(jj)2371 5306 y Fp(\034)t(;\016)o(;\027)2519 5291 y Fk(jj)p Fo(G)p Ft(\()p Fr(t)p Ft(\))p Fk(jj)2804 5306 y Fp(\034)t(;\016)o(;\027)2953 5291 y Fr(:)75 5544 y Ft(The)30 b(follo)m(wing)i(lemma)f(will)f(b)s(e)g(usefull.)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 3764 100 a Ft(14)75 399 y Fo(Lemma)35 b(6)45 b Fh(L)-5 b(et)44 b Fo(F)p Ft(\()p Fr(t)p Ft(\))g Fh(and)h Fo(G)p Ft(\()p Fr(t)p Ft(\))f Fh(b)-5 b(e)43 b(two)i(holomorphic)i(functions)c(on)h(the)g(domain)i Fr(S)3273 414 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)3464 399 y Fk([)27 b Fr(D)3627 413 y Fq(1)3710 399 y Fh(and)75 511 y(c)-5 b(ontinuous)40 b(on)691 488 y Ft(\026)675 511 y Fr(S)731 526 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)919 511 y Fk([)1026 488 y Ft(\026)1004 511 y Fr(D)1079 525 y Fq(1)1158 511 y Fh(such)f(that)h(the)g(functions)f Fr(t)d Fk(7!)h Fo(F)p Ft(\()p Fr(t)2453 478 y Fq(2)2493 511 y Ft(\))i Fh(and)h Fr(t)c Fk(7!)h Fo(G)p Ft(\()p Fr(t)3097 478 y Fq(2)3137 511 y Ft(\))i Fh(b)-5 b(elong)39 b(to)h Fk(A)3680 526 y Fp(\034)t(;\016)o(;\027)3827 511 y Fh(.)75 624 y(Then,)f(ther)-5 b(e)38 b(exists)g(a)f(function)h Fo(H)p Ft(\()p Fr(t)p Ft(\))f Fh(holomorphic)k(on)d Fr(S)2212 639 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)2398 624 y Fk([)24 b Fr(D)2558 638 y Fq(1)2635 624 y Fh(and)38 b(c)-5 b(ontinuous)38 b(on)3428 601 y Ft(\026)3413 624 y Fr(S)3469 639 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)3656 624 y Fk([)3761 601 y Ft(\026)3740 624 y Fr(D)3815 638 y Fq(1)75 748 y Fh(such)33 b(that)h Fo(H)p Ft(\()p Fr(t)618 715 y Fq(2)657 748 y Ft(\))26 b(=)814 675 y Fm(R)874 701 y Fp(t)857 780 y Fq(0)919 675 y Fm(R)980 701 y Fp(t)1005 710 y Fl(1)962 780 y Fq(0)1059 748 y Fo(F)p Ft(\(\()p Fr(t)1228 762 y Fq(1)1288 748 y Fk(\000)20 b Fr(t)1412 762 y Fq(2)1451 748 y Ft(\))1486 715 y Fq(2)1526 748 y Ft(\))p Fo(G)p Ft(\()p Fr(t)1711 715 y Fq(2)1711 773 y(2)1751 748 y Ft(\))p Fr(dt)1866 762 y Fq(2)1906 748 y Fr(dt)1986 762 y Fq(1)2058 748 y Fh(b)-5 b(elong)33 b(to)g Fk(A)2514 763 y Fp(\034)t(;\016)o(;\027)2661 748 y Fh(.)75 1010 y Fo(Pro)s(of)j Ft(T)-8 b(aking)37 b(the)e (parametrization)j Fr(t)1535 1024 y Fq(2)1608 1010 y Ft(=)c Fr(s)1756 1024 y Fq(2)1795 1010 y Fr(t)1828 1024 y Fq(1)1903 1010 y Ft(for)h Fr(s)2090 1024 y Fq(2)2163 1010 y Fk(2)f Ft([0)p Fr(;)15 b Ft(1])37 b(and)e Fr(t)2690 1024 y Fq(1)2763 1010 y Ft(=)f Fr(s)2911 1024 y Fq(1)2950 1010 y Fr(t)h Ft(for)h Fr(s)3206 1024 y Fq(1)3279 1010 y Fk(2)d Ft([0)p Fr(;)15 b Ft(1],)39 b(in)c(the)75 1123 y(in)m(tegral)d(w)m(e)f(get)397 1257 y Fm(Z)487 1284 y Fp(t)447 1463 y Fq(0)532 1257 y Fm(Z)623 1284 y Fp(t)648 1293 y Fl(1)583 1463 y Fq(0)703 1381 y Fo(F)p Ft(\(\()p Fr(t)872 1395 y Fq(1)932 1381 y Fk(\000)19 b Fr(t)1055 1395 y Fq(2)1095 1381 y Ft(\))1130 1343 y Fq(2)1169 1381 y Ft(\))p Fo(G)p Ft(\()p Fr(t)1354 1343 y Fq(2)1354 1403 y(2)1395 1381 y Ft(\))p Fr(dt)1510 1395 y Fq(2)1550 1381 y Fr(dt)1630 1395 y Fq(1)1694 1381 y Ft(=)25 b Fr(t)1823 1343 y Fq(2)1878 1257 y Fm(Z)1969 1284 y Fq(1)1928 1463 y(0)2023 1257 y Fm(Z)2114 1284 y Fq(1)2074 1463 y(0)2169 1381 y Fr(s)2212 1395 y Fq(1)2251 1381 y Fr(F)13 b Ft(\()p Fr(t)2390 1343 y Fq(2)2430 1381 y Fr(s)2473 1343 y Fq(2)2473 1403 y(1)2512 1381 y Ft(\(1)21 b Fk(\000)e Fr(s)2746 1395 y Fq(2)2786 1381 y Ft(\))2821 1343 y Fq(2)2860 1381 y Ft(\))p Fr(G)p Ft(\()p Fr(t)3034 1343 y Fq(2)3075 1381 y Fr(s)3118 1343 y Fq(2)3118 1403 y(1)3157 1381 y Fr(s)3200 1343 y Fq(2)3200 1403 y(2)3239 1381 y Ft(\))p Fr(ds)3364 1395 y Fq(2)3404 1381 y Fr(ds)3494 1395 y Fq(1)75 1631 y Ft(So)30 b(that)h(w)m(e)g(can)g(c)m(ho)s(ose)g(the)g(function)f Fo(H)p Ft(\()p Fr(t)p Ft(\))g(to)i(b)s(e)987 1894 y Fo(H)p Ft(\()p Fr(t)p Ft(\))25 b(=)g Fr(t)1341 1770 y Fm(Z)1432 1796 y Fq(1)1392 1976 y(0)1487 1770 y Fm(Z)1578 1796 y Fq(1)1537 1976 y(0)1632 1894 y Fr(s)1675 1908 y Fq(1)1714 1894 y Fr(F)13 b Ft(\()p Fr(ts)1896 1856 y Fq(2)1896 1916 y(1)1936 1894 y Ft(\(1)21 b Fk(\000)f Fr(s)2171 1908 y Fq(2)2210 1894 y Ft(\))2245 1856 y Fq(2)2284 1894 y Ft(\))p Fr(G)p Ft(\()p Fr(ts)2501 1856 y Fq(2)2501 1916 y(1)2541 1894 y Fr(s)2584 1856 y Fq(2)2584 1916 y(2)2623 1894 y Ft(\))p Fr(ds)2748 1908 y Fq(2)2788 1894 y Fr(ds)2878 1908 y Fq(1)2918 1894 y Fr(:)75 2152 y Ft(The)30 b(fact)h(that)g Fr(t)25 b Fk(7!)g Fo(H)p Ft(\()p Fr(t)962 2119 y Fq(2)1002 2152 y Ft(\))30 b(b)s(elong)h(to)g Fk(A)1542 2167 y Fp(\034)t(;\016)o(;\027)1719 2152 y Ft(follo)m(ws)h(from)e (Lemma)g(5.)1129 b Fg(2)216 2314 y Ft(W)-8 b(e)32 b(consider)e(the)h (follo)m(wing)g(in)m(tegro-di\013eren)m(tial)j(equation)75 2600 y(\(37\))367 b Fr(@)655 2562 y Fp(S)650 2622 y(z)706 2600 y Fo(W)q Ft(\()p Fr(t)883 2562 y Fq(2)923 2600 y Fr(;)15 b(z)t Ft(\))26 b(=)1178 2486 y Fp(S)t Fn(\000)p Fq(1)1181 2513 y Fm(X)1166 2711 y Fp(k)1203 2720 y Fl(1)1237 2711 y Fq(=0)1343 2600 y Fo(h)1401 2615 y Fp(k)1438 2624 y Fl(1)1476 2600 y Ft(\()p Fr(t)1544 2562 y Fq(2)1584 2600 y Ft(\))1634 2476 y Fm(Z)1725 2502 y Fp(t)1685 2682 y Fq(0)1770 2476 y Fm(Z)1861 2502 y Fp(t)1886 2511 y Fl(1)1821 2682 y Fq(0)1940 2600 y Fo(H)2022 2615 y Fp(k)2059 2624 y Fl(1)2098 2600 y Ft(\(\()p Fr(t)2201 2614 y Fq(1)2261 2600 y Fk(\000)20 b Fr(t)2385 2614 y Fq(2)2424 2600 y Ft(\))2459 2562 y Fq(2)2499 2600 y Fr(;)15 b(z)t Ft(\))p Fr(@)2673 2562 y Fp(k)2710 2571 y Fl(1)2668 2622 y Fp(z)2750 2600 y Fo(W)q Ft(\()p Fr(t)2927 2562 y Fq(2)2927 2622 y(2)2967 2600 y Fr(;)g(z)t Ft(\))p Fr(dt)3168 2614 y Fq(2)3208 2600 y Fr(dt)3288 2614 y Fq(1)75 2892 y Ft(with)30 b(initial)i(conditions)75 3096 y(\(38\))908 b(\()p Fr(@)1231 3059 y Fp(j)1226 3119 y(z)1269 3096 y Fo(W)q Ft(\)\()p Fr(t)1481 3059 y Fq(2)1521 3096 y Fr(;)15 b Ft(0\))27 b(=)e Fo(W)1872 3110 y Fp(j)1908 3096 y Ft(\()p Fr(t)1976 3059 y Fq(2)2016 3096 y Ft(\))61 b Fr(;)76 b Ft(0)25 b Fk(\024)g Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 3301 y Fh(We)32 b(assume)h(that)75 3463 y Fo(\(h1\))g Ft(F)-8 b(or)34 b(all)h(0)c Fk(\024)f Fr(j)36 b Fk(\024)31 b Fr(S)c Fk(\000)22 b Ft(1,)35 b(the)f(function)f Fo(W)1856 3477 y Fp(j)1893 3463 y Ft(\()p Fr(t)p Ft(\))h(is)f 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Ft(\))p Fr(z)2460 3846 y Fp(q)2498 3883 y Fr(;)75 4151 y Ft(are)i(con)m(v)m(ergen)m(t)h (series)f(in)f Fr(z)k Ft(on)c(a)h(disc)f Fr(D)1542 4165 y Fp(R)1627 4151 y Ft(with)g(co)s(e\016cien)m(ts)i Fr(t)c Fk(7!)h Fo(H)2545 4166 y Fp(k)2582 4175 y Fl(1)2616 4166 y Fp(;q)2673 4151 y Ft(\()p Fr(t)2741 4118 y Fq(2)2781 4151 y Ft(\))i(that)f(b)s(elong)h(to)g Fk(A)3506 4166 y Fp(\034)t(;\016)o(;\027)3680 4151 y Ft(suc)m(h)75 4264 y(that)j Fo(H)354 4279 y Fp(k)391 4288 y Fl(1)425 4279 y Fp(;q)483 4264 y Ft(\()p Fr(t)p Ft(\))g(are)f(holomorphic)h (functions)f(on)g Fr(S)1863 4279 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)2046 4264 y Fk([)20 b Fr(D)2202 4278 y Fq(1)2272 4264 y Ft(and)30 b(con)m(tin)m(uous)h(on)3045 4241 y(\026)3030 4264 y Fr(S)3086 4279 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)3269 4264 y Fk([)3371 4241 y Ft(\026)3350 4264 y Fr(D)3425 4278 y Fq(1)3465 4264 y Ft(.)75 4377 y Fo(\(h3\))f Ft(F)-8 b(or)31 b(all)h(0)26 b Fk(\024)f Fr(k)801 4391 y Fq(1)866 4377 y Fk(\024)g Fr(S)h Fk(\000)20 b Ft(1,)31 b(the)f(functions)g Fo(h)1842 4392 y Fp(k)1879 4401 y Fl(1)1918 4377 y Ft(\()p Fr(t)p Ft(\))h(are)g(holomorphic)f (functions)h(on)f Fr(S)3299 4392 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)3482 4377 y Fk([)20 b Fr(D)3638 4391 y Fq(1)3708 4377 y Ft(and)75 4490 y(con)m(tin)m(uous)26 b(on)661 4467 y(\026)646 4490 y Fr(S)702 4505 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)874 4490 y Fk([)966 4467 y Ft(\026)945 4490 y Fr(D)1020 4504 y Fq(1)1084 4490 y Ft(suc)m(h)f(that)h Fr(t)f Fk(7!)g Fo(h)1708 4505 y Fp(k)1745 4514 y Fl(1)1783 4490 y Ft(\()p Fr(t)1851 4457 y Fq(2)1891 4490 y Ft(\))g(are)h(b)s(ounded)d(holomorphic)i(functions)f (on)3502 4467 y(\026)3486 4490 y Fr(S)3542 4505 y Fp(\034)t(;\016)3644 4490 y Fk([)3736 4467 y Ft(\026)3715 4490 y Fr(D)3790 4504 y Fq(1)3829 4490 y Ft(.)75 4653 y(Using)31 b(Lemma)f(5)h(and)f (lemma)h(6,)g(the)f(same)h(pro)s(of)e(as)i(Lemma)g(4)f(yields)75 4840 y Fo(Lemma)35 b(7)45 b Fh(F)-7 b(or)36 b(given)e(initial)h(data)h Fo(W)1535 4854 y Fp(j)1572 4840 y Ft(\()p Fr(t)1640 4807 y Fq(2)1679 4840 y Ft(\))29 b Fk(2)g(A)1906 4855 y Fp(\034)t(;\016)o (;\027)2053 4840 y Fh(,)34 b Ft(0)c Fk(\024)e Fr(j)34 b Fk(\024)29 b Fr(S)d Fk(\000)c Ft(1)p Fh(,)35 b(ther)-5 b(e)35 b(exists)g(a)g(unique)f(formal)75 4953 y(series)1460 5124 y Fo(W)q Ft(\()p Fr(t)1637 5087 y Fq(2)1677 5124 y Fr(;)15 b(z)t Ft(\))26 b(=)1923 5011 y Fq(+)p Fn(1)1920 5038 y Fm(X)1922 5235 y Fp(k)r Fq(=0)2067 5124 y Fo(W)2175 5139 y Fp(k)2218 5124 y Ft(\()p Fr(t)2286 5087 y Fq(2)2325 5124 y Ft(\))2370 5063 y Fr(z)2416 5030 y Fp(k)p 2370 5103 89 4 v 2377 5187 a Fr(k)s Ft(!)75 5375 y Fh(solution)39 b(of)f(\(37\))h(and)g(\(38\))g(wher)-5 b(e)39 b Fo(W)1498 5390 y Fp(k)1540 5375 y Ft(\()p Fr(t)p Ft(\))p Fh(,)h Fr(k)e Fk(\025)c Ft(0)p Fh(,)39 b(ar)-5 b(e)39 b(holomorphic)i (functions)d(on)h Fr(S)3287 5390 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)3473 5375 y Fk([)24 b Fr(D)3633 5389 y Fq(1)3710 5375 y Fh(and)75 5488 y(c)-5 b(ontinuous)34 b(on)678 5465 y Ft(\026)663 5488 y Fr(S)719 5503 y Fq(2)p Fp(\034)t(;)p Fq(2)p Fp(\016)902 5488 y Fk([)1004 5465 y Ft(\026)983 5488 y Fr(D)1058 5502 y Fq(1)1130 5488 y Fh(such)f(that)g Fo(W)1630 5503 y Fp(k)1673 5488 y Ft(\()p Fr(t)1741 5455 y Fq(2)1781 5488 y Ft(\))25 b Fk(2)g(A)2000 5503 y Fp(\034)t(;\016)o(;\027)2147 5488 y Fh(,)33 b(for)g(al)5 b(l)33 b Fr(k)28 b Fk(\025)d Ft(0)p Fh(.)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 3764 100 a Ft(15)75 399 y(In)30 b(the)g(follo)m(wing)i(w)m(e)f(in)m(tro)s(duce)f(some)h(notations.)42 b(Let)31 b Fo(w)2170 354 y Fp(\034)t(;\016)o(;\027)2169 426 y(j)2342 399 y Ft(=)25 b Fk(jj)p Fo(W)2596 413 y Fp(j)2633 399 y Ft(\()p Fr(t)2701 366 y Fq(2)2741 399 y Ft(\))p Fk(jj)2826 414 y Fp(\034)t(;\016)o(;\027)2974 399 y Ft(,)30 b(0)c Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)31 b(and)1289 610 y Fk(j)p Fo(H)p Fk(j)1421 566 y Fp(\034)t(;\016)o(;\027)1421 640 y(k)1458 649 y Fl(1)1569 610 y Ft(\()p Fr(z)t Ft(\))26 b(=)1807 524 y Fm(X)1811 720 y Fp(q)r Fn(\025)p Fq(0)1954 610 y Fk(jj)p 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Fo(w)988 1846 y Fp(\034)t(;\016)o(;\027)987 1919 y(k)1171 1890 y Ft(=)g Fk(jj)p Fo(W)1436 1905 y Fp(k)1479 1890 y Ft(\()p Fr(t)1547 1857 y Fq(2)1587 1890 y Ft(\))p Fk(jj)1672 1905 y Fp(\034)t(;\016)o(;\027)1820 1890 y Ft(,)j(w)m(e)e(get)h(a)f (constan)m(t)h Fr(C)2702 1904 y Fq(0)2778 1890 y Fr(>)e Ft(0)h(\(one)g(can)h(c)m(ho)s(ose)f Fr(C)3708 1904 y Fq(0)3784 1890 y Ft(=)75 2003 y(max)244 2018 y Fq(0)p Fn(\024)p Fp(k)371 2027 y Fl(1)406 2018 y Fn(\024)p Fp(S)t Fn(\000)p Fq(1)617 2003 y Ft(sup)754 2026 y Fp(t)p Fn(2)838 2010 y Fq(\026)826 2026 y Fp(S)869 2038 y Fj(\034)s(;\016)954 2026 y Fn([)1017 2010 y Fq(\026)1001 2026 y Fp(D)1059 2035 y Fl(1)1112 2003 y Fk(j)p Fo(h)1195 2018 y Fp(k)1232 2027 y Fl(1)1271 2003 y Ft(\()p Fr(t)1339 1970 y Fq(2)1379 2003 y Ft(\))p Fk(j)p Ft(\),)31 b(suc)m(h)f(that)h(the)g(follo)m(wing)h (inequalities)g(hold)75 2292 y(\(39\))1041 2221 y Fo(w)1118 2176 y Fp(\034)t(;\016)o(;\027)1117 2250 y(k)r Fq(+)p Fp(S)p 1041 2271 225 4 v 1115 2355 a Fr(k)s Ft(!)1300 2292 y Fk(\024)1408 2179 y Fp(S)t Fn(\000)p Fq(1)1411 2206 y Fm(X)1396 2404 y Fp(k)1433 2413 y Fl(1)1467 2404 y Fq(=0)1647 2206 y Fm(X)1573 2404 y Fp(q)1605 2413 y Fl(1)1639 2404 y Fq(+)p Fp(q)1726 2413 y Fl(2)1760 2404 y Fq(=)p Fp(k)1868 2292 y Fr(C)1933 2306 y Fq(0)1973 2292 y Fk(jj)p Fo(H)2105 2307 y Fp(k)2142 2316 y Fl(1)2176 2307 y Fp(;q)2228 2316 y Fl(1)2266 2292 y Ft(\()p Fr(t)2334 2255 y Fq(2)2374 2292 y Ft(\))p Fk(jj)2459 2307 y Fp(\034)t(;\016)o (;\027)2617 2215 y Fo(w)2694 2170 y Fp(\034)t(;\016)o(;\027)2693 2244 y(q)2725 2253 y Fl(2)2759 2244 y Fq(+)p Fp(k)2851 2253 y Fl(1)p 2617 2271 272 4 v 2700 2355 a Fr(q)2741 2369 y Fq(2)2780 2355 y Ft(!)75 2565 y(On)e(the)g(other)h(hand,)e (consider)i(the)f(linear)h(di\013eren)m(tial)h(equation)75 2827 y(\(40\))1016 b Fr(@)1304 2790 y Fp(S)1299 2850 y(z)1356 2827 y Fo(V)q Ft(\()p Fr(z)t Ft(\))26 b(=)1687 2714 y Fp(S)t Fn(\000)p Fq(1)1689 2741 y Fm(X)1674 2939 y Fp(k)1711 2948 y Fl(1)1746 2939 y Fq(=0)1851 2827 y Fr(C)1916 2841 y Fq(0)1955 2827 y Fk(j)p Fo(H)p Fk(j)2087 2783 y Fp(\034)t(;\016)o(;\027)2087 2857 y(k)2124 2866 y Fl(1)2235 2827 y Ft(\()p Fr(z)t Ft(\))p Fr(@)2404 2790 y Fp(k)2441 2799 y Fl(1)2399 2850 y Fp(z)2481 2827 y Fo(V)q Ft(\()p Fr(z)t Ft(\))75 3096 y(with)k(initial)i(conditions)75 3277 y(\(41\))1011 b(\()p Fr(@)1334 3240 y Fp(j)1329 3300 y(z)1371 3277 y Fo(V)q Ft(\)\(0\))27 b(=)e Fo(w)1801 3233 y Fp(\034)t(;\016)o(;\027)1800 3305 y(j)2009 3277 y Fr(;)76 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 3458 y Ft(F)-8 b(rom)29 b(the)g(classical)h Fh(glob)-5 b(al)30 b Ft(theory)f(of)f(linear)h (di\013eren)m(tial)h(equations)f(in)g(the)f(complex)i(domain,)f(w)m(e)f (kno)m(w)75 3571 y(that)i(the)g(problem)g(\(40\),)h(\(41\))h(has)d(a)h (unique)f(solution)h Fo(V)q Ft(\()p Fr(z)t Ft(\))d(=)2382 3503 y Fm(P)2478 3598 y Fp(p)p Fn(\025)p Fq(0)2623 3571 y Fo(V)2702 3585 y Fp(p)2742 3571 y Fr(z)2788 3538 y Fp(p)2828 3571 y Fr(=p)p Ft(!,)j(holomorphic)g(on)g(ev)m(ery)75 3710 y(disc)44 b Fr(D)346 3725 y Fp(R)399 3706 y Fd(0)471 3710 y Ft(con)m(tained)i(in)e(the)h(op)s(en)f(maximal)h(domain)g Fk(D)2215 3725 y Fp(\034)t(;\016)o(;\027)2406 3710 y Ft(where)f(the)h(co)s(e\016cien)m(ts)h Fr(C)3393 3724 y Fq(0)3433 3710 y Fk(j)p Fo(H)p Fk(j)3565 3666 y Fp(\034)t(;\016)o (;\027)3565 3739 y(k)3602 3748 y Fl(1)3712 3710 y Ft(\()p Fr(z)t Ft(\),)75 3823 y(0)26 b Fk(\024)f Fr(k)289 3837 y Fq(1)354 3823 y Fk(\024)g Fr(S)g Fk(\000)20 b Ft(1,)31 b(are)g(holomorphic)f(functions.)216 3936 y(No)m(w,)k(a)e(formal)g (series)g Fo(V)q Ft(\()p Fr(z)t Ft(\))e(=)1385 3867 y Fm(P)1481 3962 y Fp(k)r Fn(\025)p Fq(0)1629 3936 y Fo(V)1708 3951 y Fp(k)1750 3936 y Fr(z)1796 3903 y Fp(k)1839 3936 y Fr(=k)s Ft(!)j(is)f(a)h(solution)f(of)g(\(40\))i(and)d(\(41\))j(if)e (and)f(only)h(if)g(the)75 4048 y(follo)m(wing)g(equalities)g(hold)75 4311 y(\(42\))1039 4249 y Fo(V)1118 4264 y Fp(k)r Fq(+)p Fp(S)p 1039 4290 224 4 v 1113 4373 a Fr(k)s Ft(!)1298 4311 y(=)1406 4197 y Fp(S)t Fn(\000)p Fq(1)1409 4224 y Fm(X)1394 4422 y Fp(k)1431 4431 y Fl(1)1465 4422 y Fq(=0)1645 4224 y Fm(X)1570 4422 y Fp(q)1602 4431 y Fl(1)1637 4422 y Fq(+)p Fp(q)1724 4431 y Fl(2)1757 4422 y Fq(=)p Fp(k)1866 4311 y Fr(C)1931 4325 y Fq(0)1970 4311 y Fk(jj)p Fo(H)2102 4326 y Fp(k)2139 4335 y Fl(1)2174 4326 y Fp(;q)2226 4335 y Fl(1)2264 4311 y Ft(\()p Fr(t)2332 4273 y Fq(2)2372 4311 y Ft(\))p Fk(jj)2457 4326 y Fp(\034)t(;\016)o(;\027)2615 4247 y Fo(V)2694 4262 y Fp(q)2726 4271 y Fl(2)2760 4262 y Fq(+)p Fp(k)2852 4271 y Fl(1)p 2615 4290 276 4 v 2700 4373 a Fr(q)2741 4387 y Fq(2)2780 4373 y Ft(!)75 4583 y(for)e(all)h Fr(k)e Fk(\025)c Ft(0)31 b(and)75 4764 y(\(43\))1132 b Fo(V)1446 4778 y Fp(j)1508 4764 y Ft(=)25 b Fo(w)1681 4720 y Fp(\034)t(;\016)o(;\027)1680 4792 y(j)1888 4764 y Fr(;)76 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 4945 y Ft(W)-8 b(e)32 b(deduce)e(from)f(\(39\))j(and)e(\(42\))i(with)e (\(43\),)i(that)1720 5126 y Fo(w)1797 5089 y Fp(\034)t(;\016)o(;\027) 1796 5149 y(p)1970 5126 y Fk(\024)24 b Fo(V)2144 5140 y Fp(p)2184 5126 y Fr(;)75 5307 y Ft(for)j(all)i Fr(p)c Fk(\025)g Ft(0.)40 b(So)27 b(that,)j(for)d(ev)m(ery)h Fr(R)1399 5274 y Fn(0)1448 5307 y Fr(>)c Ft(0,)29 b(suc)m(h)f(that)g Fr(D)2114 5323 y Fp(R)2167 5304 y Fd(0)2222 5307 y Ft(is)f(con)m (tained)i(in)e(the)h(op)s(en)f(maximal)i(domain)75 5420 y Fk(D)145 5435 y Fp(\034)t(;\016)o(;\027)316 5420 y Ft(where)24 b(the)g(co)s(e\016cien)m(ts)h Fr(C)1241 5434 y Fq(0)1281 5420 y Fk(j)p Fo(H)p Fk(j)1413 5376 y Fp(\034)t(;\016)o (;\027)1413 5450 y(k)1450 5459 y Fl(1)1560 5420 y Ft(\()p Fr(z)t Ft(\),)i(0)e Fk(\024)g Fr(k)1941 5434 y Fq(1)2006 5420 y Fk(\024)g Fr(S)12 b Fk(\000)7 b Ft(1,)26 b(are)f(holomorphic)e (functions,)i(w)m(e)g(get)g(that)75 5558 y(the)j(series)471 5489 y Fm(P)567 5584 y Fp(p)p Fn(\025)p Fq(0)712 5558 y Fo(w)789 5513 y Fp(\034)t(;\016)o(;\027)788 5569 y(p)936 5558 y Fr(z)982 5525 y Fp(p)1022 5558 y Fr(=p)p Ft(!)g(con)m(v)m(erges) i(on)e Fr(D)1770 5573 y Fp(R)1823 5554 y Fd(0)1879 5558 y Ft(and)f(that)i(the)f(formal)g(solution)h Fo(W)q Ft(\()p Fr(t)3204 5525 y Fq(2)3244 5558 y Fr(;)15 b(z)t Ft(\))29 b(constructed)75 5670 y(in)h(Lemma)h(7,)g(is)f(a)h(con)m(v)m(ergen)m(t) i(series)d(in)g Fr(z)35 b Ft(on)30 b Fr(D)1856 5686 y Fp(R)1909 5667 y Fd(0)1967 5670 y Ft(with)g(v)-5 b(alues)31 b(in)f Fk(A)2624 5685 y Fp(\034)t(;\016)o(;\027)2770 5670 y Ft(.)961 b Fg(2)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 3764 100 a Ft(16)75 399 y Fs(7)135 b(Hyp)t(erfunctions)44 b(solutions)i(of)g(Cauc)l(h)l(y)f(problems)g(in) g Ff(D)3255 355 y Fe(0)3252 428 y Fv(\027)75 605 y Fc(7.1)112 b(Assumptions)39 b(on)f(the)f(data)h(of)g(the)f(Cauc)m(h)m(y)h(problem) 75 776 y Ft(Let)33 b Fr(\022)286 738 y Fn(\000)283 802 y Fq(1)344 776 y Fr(;)15 b(\022)430 738 y Fq(+)427 802 y(1)521 776 y Ft(b)s(e)31 b(t)m(w)m(o)j(real)f(n)m(um)m(b)s(ers)e(suc)m (h)h(that)h(0)28 b Fr(<)g(\022)1992 738 y Fq(+)1989 802 y(1)2072 776 y Fk(\000)21 b Fr(\022)2210 738 y Fn(\000)2207 802 y Fq(1)2297 776 y Fr(<)28 b(\031)35 b Ft(and)d(let)h Fr(\022)2838 790 y Fq(1)2906 776 y Ft(=)28 b(\()p Fr(\022)3086 738 y Fq(+)3083 802 y(1)3166 776 y Ft(+)21 b Fr(\022)3304 738 y Fn(\000)3301 802 y Fq(1)3362 776 y Ft(\))p Fr(=)p Ft(2.)48 b(W)-8 b(e)33 b(put)75 889 y Fr(\034)125 856 y Fn(\000)211 889 y Ft(=)26 b(\()p Fr(\022)389 851 y Fn(\000)386 915 y Fq(1)469 889 y Ft(+)20 b Fr(\022)603 903 y Fq(1)642 889 y Ft(\))p Fr(=)p Ft(2)33 b(and)d(consider)h(an)h (increasing)f(sequence)h Fr(\016)2306 856 y Fn(\000)2303 912 y Fp(n)2393 889 y Fr(>)26 b Ft(0,)32 b Fr(n)26 b Fk(\025)h Ft(1,)32 b(suc)m(h)f(that)h Fr(\034)3327 856 y Fn(\000)3407 889 y Fk(\000)20 b Fr(\016)3541 856 y Fn(\000)3538 912 y Fp(n)3628 889 y Fr(>)26 b(\022)3771 851 y Fn(\000)3768 915 y Fq(1)3829 889 y Ft(,)75 1002 y Fr(\034)125 969 y Fn(\000)206 1002 y Ft(+)c Fr(\016)342 969 y Fn(\000)339 1025 y Fp(n)431 1002 y Fr(<)30 b(\022)575 1016 y Fq(1)647 1002 y Ft(and)i Fr(\034)876 969 y Fn(\000)958 1002 y Ft(+)21 b Fr(\016)1093 969 y Fn(\000)1090 1025 y Fp(n)1183 1002 y Fk(!)30 b Fr(\022)1347 1016 y Fq(1)1419 1002 y Ft(as)j Fr(n)c Fk(!)h Ft(+)p Fk(1)p Ft(.)48 b(W)-8 b(e)35 b(also)f(put)e Fr(\034)2537 969 y Fq(+)2626 1002 y Ft(=)d(\()p Fr(\022)2804 1016 y Fq(1)2866 1002 y Ft(+)22 b Fr(\022)3005 964 y Fq(+)3002 1028 y(1)3063 1002 y Ft(\))p Fr(=)p Ft(2)34 b(and)f(consider)g(an)75 1115 y(increasing)d(sequence)g Fr(\016)918 1082 y Fq(+)915 1138 y Fp(n)1003 1115 y Fr(>)25 b Ft(0,)30 b Fr(n)25 b Fk(\025)g Ft(1,)30 b(suc)m(h)f(that)h Fr(\034)1925 1082 y Fq(+)2002 1115 y Ft(+)18 b Fr(\016)2134 1082 y Fn(\000)2131 1138 y Fp(n)2219 1115 y Fr(<)25 b(\022)2361 1077 y Fq(+)2358 1141 y(1)2419 1115 y Ft(,)30 b Fr(\034)2524 1082 y Fq(+)2601 1115 y Fk(\000)18 b Fr(\016)2733 1082 y Fq(+)2730 1138 y Fp(n)2818 1115 y Fr(>)25 b(\022)2957 1129 y Fq(1)3026 1115 y Ft(and)j Fr(\034)3251 1082 y Fq(+)3329 1115 y Fk(\000)18 b Fr(\016)3461 1082 y Fq(+)3458 1138 y Fp(n)3546 1115 y Fk(!)25 b Fr(\022)3705 1129 y Fq(1)3773 1115 y Ft(as)75 1228 y Fr(n)g Fk(!)g Ft(+)p Fk(1)p Ft(.)40 b(W)-8 b(e)32 b(also)f(in)m(tro)s(duce)f(the)h(follo)m (wing)h(sets)f(in)f Fi(C)870 1429 y Fk(W)j Ft(=)25 b Fk(f)p Fr(t)g Fk(2)g Fi(C)1344 1392 y Fn(\003)1383 1429 y Fr(=)15 b Ft(arg)s(\()p Fr(t)p Ft(\))26 b Fk(2)e Ft(\()p Fr(\022)1867 1391 y Fn(\000)1864 1455 y Fq(1)1926 1429 y Fr(;)15 b(\022)2012 1391 y Fq(+)2009 1455 y(1)2071 1429 y Ft(\))p Fr(;)76 b(t)25 b Fk(6)p Ft(=)g Fr(k)s(e)2453 1392 y Fp(i\022)2510 1401 y Fl(1)2549 1429 y Fr(;)107 b Fk(8)p Fr(k)28 b Fk(2)c Fi(N)30 b Fk(g)p Fr(;)537 1630 y Fk(W)634 1593 y Fq(+)718 1630 y Ft(=)25 b Fk(f)p Fr(t)h Fk(2)f(W)7 b Fr(=)15 b Ft(arg)r(\()p Fr(t)p Ft(\))26 b Fk(2)f Ft(\()p Fr(\022)1582 1644 y Fq(1)1622 1630 y Fr(;)15 b(\022)1705 1644 y Fq(1)1744 1630 y Ft(+\))p Fk(g)61 b Fr(;)76 b Fk(W)2154 1593 y Fn(\000)2239 1630 y Ft(=)25 b Fk(f)p Fr(t)g Fk(2)g(W)7 b Fr(=)15 b Ft(arg)s(\()p Fr(t)p Ft(\))26 b Fk(2)e Ft(\()p Fr(\022)3105 1592 y Fn(\000)3102 1656 y Fq(1)3164 1630 y Fr(;)15 b(\022)3247 1644 y Fq(1)3287 1630 y Ft(\))p Fk(g)p Fr(:)75 1795 y Ft(W)-8 b(e)32 b(consider)e(the)g(follo)m(wing)i(Cauc)m(h)m(y)f (problems)75 2078 y(\(44\))308 b Fr(@)596 2040 y Fp(S)591 2100 y(z)647 2078 y Fo(W)756 2040 y Fn(\000)815 2078 y Ft(\()p Fr(t)883 2040 y Fq(2)923 2078 y Fr(;)15 b(z)t Ft(\))26 b(=)1178 1964 y Fp(S)t Fn(\000)p Fq(1)1181 1991 y Fm(X)1166 2189 y Fp(k)1203 2198 y Fl(1)1237 2189 y Fq(=0)1343 2078 y Fo(h)1401 2093 y Fp(k)1438 2102 y Fl(1)1476 2078 y Ft(\()p Fr(t)1544 2040 y Fq(2)1584 2078 y Ft(\))1634 1954 y Fm(Z)1725 1980 y Fp(t)1685 2160 y Fq(0)1770 1954 y Fm(Z)1861 1980 y Fp(t)1886 1989 y Fl(1)1821 2160 y Fq(0)1940 2078 y Fo(H)2022 2039 y Fn(\000)2022 2107 y Fp(k)2059 2116 y Fl(1)2098 2078 y Ft(\(\()p Fr(t)2201 2092 y Fq(1)2261 2078 y Fk(\000)20 b Fr(t)2385 2092 y Fq(2)2424 2078 y Ft(\))2459 2040 y Fq(2)2499 2078 y Fr(;)15 b(z)t Ft(\))p Fr(@)2673 2040 y Fp(k)2710 2049 y Fl(1)2668 2100 y Fp(z)2750 2078 y Fo(W)2859 2040 y Fn(\000)2918 2078 y Ft(\()p Fr(t)2986 2040 y Fq(2)2986 2100 y(2)3026 2078 y Fr(;)g(z)t Ft(\))p Fr(dt)3227 2092 y Fq(2)3267 2078 y Fr(dt)3347 2092 y Fq(1)75 2367 y Ft(with)30 b(initial)i (conditions)75 2568 y(\(45\))867 b(\()p Fr(@)1190 2530 y Fp(j)1185 2590 y(z)1227 2568 y Fo(W)1336 2530 y Fn(\000)1396 2568 y Ft(\)\()p Fr(t)1499 2530 y Fq(2)1539 2568 y Fr(;)15 b Ft(0\))26 b(=)f Fo(W)1890 2530 y Fn(\000)1889 2595 y Fp(j)1950 2568 y Ft(\()p Fr(t)2018 2530 y Fq(2)2057 2568 y Ft(\))61 b Fr(;)76 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 2769 y Ft(and)75 3033 y(\(46\))308 b Fr(@)596 2996 y Fp(S)591 3056 y(z)647 3033 y Fo(W)756 2996 y Fq(+)815 3033 y Ft(\()p Fr(t)883 2996 y Fq(2)923 3033 y Fr(;)15 b(z)t Ft(\))26 b(=)1178 2920 y Fp(S)t Fn(\000)p Fq(1)1181 2947 y Fm(X)1166 3145 y Fp(k)1203 3154 y Fl(1)1237 3145 y Fq(=0)1343 3033 y Fo(h)1401 3048 y Fp(k)1438 3057 y Fl(1)1476 3033 y Ft(\()p Fr(t)1544 2996 y Fq(2)1584 3033 y Ft(\))1634 2910 y Fm(Z)1725 2936 y Fp(t)1685 3116 y Fq(0)1770 2910 y Fm(Z)1861 2936 y Fp(t)1886 2945 y 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4012 y Fr(S)3525 4043 y Fq(2)p Fp(\034)3599 4024 y Fl(+)3649 4043 y Fp(;)p Fq(2)p Fp(\016)3737 4013 y Fl(+)3735 4053 y Fj(n)3794 4012 y Fk([)96 4119 y Ft(\026)75 4142 y Fr(D)150 4156 y Fq(1)190 4142 y Ft(\))33 b(for)f(all)i Fr(n)28 b Fk(\025)h Ft(1.)48 b(Moreo)m(v)m(er,)35 b(for)e(0)c Fk(\024)g Fr(j)35 b Fk(\024)28 b Fr(S)f Fk(\000)21 b Ft(1,)34 b(all)g Fr(n)28 b Fk(\025)h Ft(1,)34 b(the)f(function)f Fr(t)d Fk(7!)g Fo(W)3246 4104 y Fn(\000)3245 4169 y Fp(j)3306 4142 y Ft(\()p Fr(t)3374 4109 y Fq(2)3413 4142 y Ft(\))k(b)s(elong)g (to)75 4269 y Fk(A)148 4299 y Fp(\034)187 4280 y Fd(\000)239 4299 y Fp(;\016)292 4270 y Fd(\000)290 4310 y Fj(n)344 4299 y Fp(;\027)441 4269 y Ft(and)h(the)h(function)f Fr(t)e Fk(7!)h Fo(W)1441 4231 y Fq(+)1440 4296 y Fp(j)1500 4269 y Ft(\()p Fr(t)1568 4236 y Fq(2)1608 4269 y Ft(\))i(b)s(elong)f (to)i Fk(A)2161 4299 y Fp(\034)2200 4280 y Fl(+)2250 4299 y Fp(;\016)2303 4270 y Fl(+)2301 4310 y Fj(n)2354 4299 y Fp(;\027)2417 4269 y Ft(.)53 b(W)-8 b(e)36 b(assume)e(that)h Fo(W)3284 4231 y Fn(\000)3283 4296 y Fp(j)3344 4269 y Ft(\()p Fr(t)p Ft(\))d(=)g Fo(W)3691 4231 y Fq(+)3690 4296 y Fp(j)3751 4269 y Ft(\()p Fr(t)p Ft(\))75 4382 y(for)e(all)h Fr(t)25 b Fk(2)g Fr(D)559 4396 y Fq(1)599 4382 y Ft(,)31 b(for)f(all)h(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1.)216 4495 y(F)-8 b(or)28 b(ev)m(ery)f(\014xed)f Fr(\022)i Fk(2)d Ft(\()p Fr(\022)1066 4457 y Fn(\000)1063 4521 y Fq(1)1124 4495 y Fr(;)15 b(\022)1207 4509 y Fq(1)1247 4495 y Ft(\),)28 b(consider)f(the)f(function)h Fr(W)2290 4457 y Fn(\000)2277 4524 y Fp(j;\022)2364 4495 y Ft(\()p Fr(r)s Ft(\))e(=)g Fo(W)2708 4457 y Fn(\000)2707 4522 y Fp(j)2768 4495 y Ft(\()p Fr(r)2847 4462 y Fq(2)2886 4495 y Fr(e)2928 4462 y Fq(2)p Fp(i\022)3027 4495 y Ft(\))i(de\014ned)f(on)g([0)p Fr(;)15 b Ft(+)p Fk(1)p Ft(\).)75 4624 y(As)32 b(a)h(result,)f(the)h (function)e Fr(r)g Fk(7!)d Fr(W)1375 4586 y Fn(\000)1362 4653 y Fp(j;\022)1449 4624 y Ft(\()p Fr(r)s Ft(\))k(is)h(a)f(con)m(tin) m(uous)h(function)f(from)f([0)p Fr(;)15 b Ft(+)p Fk(1)p Ft(\))33 b(in)m(to)h Fi(C)p Ft(.)45 b(Moreo)m(v)m(er,)75 4737 y(w)m(e)31 b(ha)m(v)m(e)1479 4850 y(sup)1421 4932 y Fp(r)r Fn(2)p Fq([0)p Fp(;)p Fn(1)p Fq(\))1689 4850 y Fk(j)p Fr(W)1813 4811 y Fn(\000)1800 4879 y Fp(j;\022)1887 4850 y Ft(\()p Fr(r)s Ft(\))p Fk(j)p Fr(e)2068 4812 y Fn(\000)p Fp(\027)t(r)2226 4850 y Fr(<)25 b Ft(+)p Fk(1)p Fr(:)75 5097 y Ft(In)k(particular,)i(from)e(the)h(prop)s(osition)g(2,)h (the)f(function)f Fr(r)f Fk(7!)d Fr(W)2377 5059 y Fn(\000)2364 5126 y Fp(j;\022)2451 5097 y Ft(\()p Fr(r)s Ft(\))30 b(b)s(elong)g(to)h(the)f(Banac)m(h)h(space)g Fk(D)3792 5064 y Fn(0)3789 5125 y Fp(\027)3828 5106 y Fd(0)75 5210 y Ft(for)f(an)m(y)h Fr(\027)437 5177 y Fn(0)485 5210 y Fr(>)25 b(\027)6 b Ft(.)216 5323 y(F)-8 b(or)28 b(ev)m(ery)f(\014xed) f Fr(\022)i Fk(2)d Ft(\()p Fr(\022)1063 5337 y Fq(1)1102 5323 y Fr(;)15 b(\022)1188 5285 y Fq(+)1185 5349 y(1)1247 5323 y Ft(\),)28 b(consider)f(the)f(function)h Fr(W)2290 5285 y Fq(+)2277 5352 y Fp(j;\022)2364 5323 y Ft(\()p Fr(r)s Ft(\))e(=)g Fo(W)2708 5285 y Fq(+)2707 5350 y Fp(j)2768 5323 y Ft(\()p Fr(r)2847 5290 y Fq(2)2886 5323 y Fr(e)2928 5290 y Fp(i)p Fq(2)p Fp(\022)3027 5323 y Ft(\))i(de\014ned)f(on)g([0)p Fr(;)15 b Ft(+)p Fk(1)p Ft(\).)75 5452 y(As)32 b(a)h(result,)f(the)h(function)e Fr(r)g Fk(7!)d Fr(W)1375 5414 y Fq(+)1362 5481 y Fp(j;\022)1449 5452 y Ft(\()p Fr(r)s Ft(\))k(is)h(a)f(con)m(tin)m(uous)h(function)f (from)f([0)p Fr(;)15 b Ft(+)p Fk(1)p Ft(\))33 b(in)m(to)h Fi(C)p Ft(.)45 b(Moreo)m(v)m(er,)75 5565 y(w)m(e)31 b(ha)m(v)m(e)1479 5678 y(sup)1421 5760 y Fp(r)r Fn(2)p Fq([0)p Fp(;)p Fn(1)p Fq(\))1689 5678 y Fk(j)p Fr(W)1813 5639 y Fq(+)1800 5707 y Fp(j;\022)1887 5678 y Ft(\()p Fr(r)s Ft(\))p Fk(j)p Fr(e)2068 5640 y Fn(\000)p Fp(\027)t(r)2226 5678 y Fr(<)25 b Ft(+)p Fk(1)p Fr(:)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 3764 100 a Ft(17)75 399 y(In)29 b(particular,)i(from)e(the)h(prop)s(osition)g(2,)h(the)f(function)f Fr(r)f Fk(7!)d Fr(W)2377 360 y Fq(+)2364 428 y Fp(j;\022)2451 399 y Ft(\()p Fr(r)s Ft(\))30 b(b)s(elong)g(to)h(the)f(Banac)m(h)h (space)g Fk(D)3792 366 y Fn(0)3789 427 y Fp(\027)3828 408 y Fd(0)75 511 y Ft(for)f(an)m(y)h Fr(\027)437 478 y Fn(0)485 511 y Fr(>)25 b(\027)6 b Ft(.)40 b(W)-8 b(e)32 b(mak)m(e)f(the)g(follo)m(wing)h(assumption.)216 624 y(W)-8 b(e)27 b(tak)m(e)g Fr(\027)611 591 y Fn(0)660 624 y Fr(>)e(\027)31 b Ft(suc)m(h)25 b(that)i Fr(\027)1276 591 y Fn(0)1299 624 y Fr(e)1341 591 y Fq(1)p Fn(\000)p Fp(\027)1470 568 y Fd(0)1522 624 y Fr(<)e Ft(1.)40 b(There)25 b(exist)h(staircase)i(distributions)d Fr(W)3188 586 y Fn(\000)3175 654 y Fp(j;\022)3257 663 y Fl(1)3294 624 y Ft(\()p Fr(r)s Ft(\))p Fr(;)15 b(W)3547 586 y Fq(+)3534 654 y Fp(j;\022)3616 663 y Fl(1)3654 624 y Ft(\()p Fr(r)s Ft(\))26 b Fk(2)75 737 y(D)148 704 y Fn(0)145 765 y Fp(\027)184 746 y Fd(0)241 737 y Ft(suc)m(h)k(that)75 850 y Fo(a\))1340 963 y Ft(lim)1229 1024 y Fp(\022)r Fn(!)p Fp(\022)1368 1033 y Fl(1)1402 1024 y Fp(;\022)r(<\022)1545 1033 y Fl(1)1594 963 y Fk(jj)p Fr(W)1743 925 y Fn(\000)1730 992 y Fp(j;\022)1817 963 y Ft(\()p Fr(r)s Ft(\))21 b Fk(\000)e Fr(W)2141 925 y Fn(\000)2128 992 y Fp(j;\022)2210 1001 y Fl(1)2248 963 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2412 978 y Fp(\027)2451 959 y Fd(0)2474 978 y Fp(;d)2559 963 y Ft(=)25 b(0)75 1170 y Fo(b\))1328 1283 y Ft(lim)1216 1344 y Fp(\022)r Fn(!)p Fp(\022)1355 1353 y Fl(1)1389 1344 y Fp(;\022)r(>\022)1532 1353 y Fl(1)1581 1283 y Fk(jj)p Fr(W)1730 1245 y Fq(+)1717 1312 y Fp(j;\022)1804 1283 y Ft(\()p Fr(r)s Ft(\))c Fk(\000)f Fr(W)2129 1245 y Fq(+)2116 1312 y Fp(j;\022)2198 1321 y Fl(1)2235 1283 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2399 1298 y Fp(\027)2438 1279 y Fd(0)2462 1298 y Fp(;d)2547 1283 y Ft(=)25 b(0)p Fr(:)75 1500 y Ft(In)36 b(particular,)j(from)d(the)g(prop)s(osition)h (6,)h(for)f(0)f Fk(\024)f Fr(j)41 b Fk(\024)36 b Fr(S)29 b Fk(\000)24 b Ft(1,)39 b(the)e(distribution)f Fr(W)3195 1462 y Fq(+)3182 1529 y Fp(j;\022)3264 1538 y Fl(1)3301 1500 y Ft(\()p Fr(r)s Ft(\))25 b Fk(\000)f Fr(W)3634 1462 y Fn(\000)3621 1529 y Fp(j;\022)3703 1538 y Fl(1)3740 1500 y Ft(\()p Fr(r)s Ft(\))75 1630 y(de\014nes)29 b(a)i(h)m(yp)s (erfunction)e(on)i(the)f(half-line)h Fi(R)1730 1644 y Fq(+)1789 1630 y Fr(e)1831 1597 y Fp(i\022)1888 1606 y Fl(1)1952 1630 y Ft(=)25 b Fk(f)p Fr(t)h Fk(2)e Fi(C)2303 1597 y Fn(\003)2368 1630 y Ft(:)h Fr(t)g Ft(=)g Fr(r)s(e)2658 1597 y Fp(i\022)2715 1606 y Fl(1)2754 1630 y Fr(;)15 b(r)28 b(>)d Ft(0)p Fk(g)p Ft(.)75 1788 y Fo(\(h2'\))30 b Ft(F)-8 b(or)31 b(all)g(0)26 b Fk(\024)f Fr(k)829 1802 y Fq(1)894 1788 y Fk(\024)g Fr(S)g Fk(\000)20 b Ft(1,)1418 1914 y Fo(H)1500 1876 y Fn(\000)1500 1944 y Fp(k)1537 1953 y Fl(1)1576 1914 y Ft(\()p Fr(t)1644 1877 y Fq(2)1683 1914 y Fr(;)15 b(z)t Ft(\))27 b(=)1927 1828 y Fm(X)1930 2024 y Fp(q)r Fn(\025)p Fq(0)2073 1914 y Fo(H)2155 1876 y Fn(\000)2155 1944 y Fp(k)2192 1953 y Fl(1)2226 1944 y Fp(;q)2284 1914 y Ft(\()p Fr(t)2352 1877 y Fq(2)2391 1914 y Ft(\))p Fr(z)2472 1877 y Fp(q)75 2180 y Ft(are)34 b(con)m(v)m(ergen)m(t)j(series)d(in)g Fr(z)k Ft(on)c(a)g(disc)g Fr(D)1596 2194 y Fp(R)1649 2203 y Fl(1)1722 2180 y Ft(with)g(co)s (e\016cien)m(ts)h Fr(t)d Fk(7!)f Fo(H)2665 2141 y Fn(\000)2665 2209 y Fp(k)2702 2218 y Fl(1)2736 2209 y Fp(;q)2794 2180 y Ft(\()p Fr(t)2862 2147 y Fq(2)2901 2180 y Ft(\),)36 b Fr(q)e Fk(\025)d Ft(0,)k(that)g(b)s(elong)f(to)75 2309 y Fk(A)148 2339 y Fp(\034)187 2320 y Fd(\000)239 2339 y Fp(;\016)292 2309 y Fd(\000)290 2349 y Fj(n)344 2339 y Fp(;\027)446 2309 y Ft(suc)m(h)39 b(that)h(the)g(function)f Fo(H)1479 2270 y Fn(\000)1479 2338 y Fp(k)1516 2347 y Fl(1)1550 2338 y Fp(;q)1607 2309 y Ft(\()p Fr(t)p Ft(\))h(is)f (holomorphic)h(on)f Fr(S)2571 2339 y Fq(2)p Fp(\034)2645 2320 y Fd(\000)2697 2339 y Fp(;)p Fq(2)p Fp(\016)2785 2309 y Fd(\000)2783 2349 y Fj(n)2868 2309 y Fk([)26 b Fr(D)3030 2323 y Fq(1)3109 2309 y Ft(and)39 b(con)m(tin)m(uous)h(on)90 2412 y(\026)75 2435 y Fr(S)131 2465 y Fq(2)p Fp(\034)205 2446 y Fd(\000)257 2465 y Fp(;)p Fq(2)p 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b(mak)m(e)h(the)e(follo)m(wing)i(assumption)1462 4068 y Fr(D)1537 4082 y Fp(R)1590 4091 y Fl(1)1655 4068 y Fk(\032)1767 3981 y Fm(\\)1751 4177 y Fp(n)p Fn(\025)p Fq(1)1899 4068 y Fk(D)1969 4098 y Fp(\034)2008 4079 y Fd(\000)2060 4098 y Fp(;\034)2119 4079 y Fl(+)2170 4098 y Fp(;\016)2223 4068 y Fd(\000)2221 4108 y Fj(n)2275 4098 y Fp(;\016)2328 4068 y Fl(+)2326 4108 y Fj(n)2379 4098 y Fp(;\027)2442 4068 y Fr(:)75 4362 y Ft(F)-8 b(or)31 b(ev)m(ery)g(\014xed)f Fr(\022)d Fk(2)e Ft(\()p Fr(\022)935 4324 y Fn(\000)932 4388 y Fq(1)994 4362 y Fr(;)15 b(\022)1077 4376 y Fq(1)1116 4362 y Ft(\),)31 b(consider)g(the)f(function)g Fr(H)2157 4324 y Fn(\000)2150 4392 y Fp(k)2187 4401 y Fl(1)2221 4392 y Fp(;\022)2280 4362 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)e Fo(H)2679 4324 y Fn(\000)2679 4392 y Fp(k)2716 4401 y Fl(1)2754 4362 y Ft(\()p Fr(r)2833 4329 y Fq(2)2872 4362 y Fr(e)2914 4329 y Fq(2)p Fp(i\022)3013 4362 y Fr(;)15 b(z)t Ft(\))32 b(with)e(expansion)1389 4586 y Fr(H)1472 4548 y Fn(\000)1465 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5137 y Fp(k)2289 5146 y Fl(1)2323 5137 y Fp(;\022)2381 5108 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))28 b(=)d Fo(H)2781 5070 y Fq(+)2781 5137 y Fp(k)2818 5146 y Fl(1)2856 5108 y Ft(\()p Fr(r)2935 5075 y Fq(2)2974 5108 y Fr(e)3016 5075 y Fq(2)p Fp(i\022)3115 5108 y Fr(;)15 b(z)t Ft(\))25 b(with)g(expansion)1389 5331 y Fr(H)1472 5293 y Fq(+)1465 5361 y Fp(k)1502 5370 y Fl(1)1536 5361 y Fp(;\022)1594 5331 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)1911 5245 y Fm(X)1915 5441 y Fp(q)r Fn(\025)p Fq(0)2058 5331 y Fr(H)2141 5293 y Fq(+)2134 5361 y Fp(k)2171 5370 y Fl(1)2205 5361 y Fp(;\022)r(;q)2317 5331 y Ft(\()p Fr(r)s Ft(\))p Fr(z)2477 5294 y Fp(q)2515 5331 y Fr(:)75 5627 y Ft(As)36 b(a)h(result,)h(from)e(the)h(prop)s (osition)f(2,)j Fr(H)1643 5589 y Fq(+)1636 5657 y Fp(k)1673 5666 y Fl(1)1707 5657 y Fp(;\022)1765 5627 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))39 b(is)d(a)h(con)m(v)m(ergen)m(t)i(series)e (in)f(the)g(v)-5 b(ariable)38 b Fr(z)h Fk(2)c Fr(D)3762 5641 y Fp(R)3815 5650 y Fl(1)75 5740 y Ft(with)30 b(v)-5 b(alues)31 b(in)f Fk(D)732 5707 y Fn(0)729 5768 y Fp(\027)768 5749 y Fd(0)795 5740 y Ft(.)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 3764 100 a Ft(18)216 399 y(W)-8 b(e)51 b(mak)m(e)f(the)f(follo)m(wing)i(assumption.)96 b(There)49 b(exist)h(staircase)h(distributions)d Fr(H)3336 360 y Fn(\000)3329 428 y Fp(k)3366 437 y Fl(1)3400 428 y Fp(;\022)3453 437 y Fl(1)3487 428 y Fp(;q)3545 399 y Ft(\()p Fr(r)s Ft(\))h(and)75 527 y Fr(H)158 489 y Fq(+)151 557 y Fp(k)188 566 y Fl(1)222 557 y Fp(;\022)275 566 y Fl(1)309 557 y Fp(;q)367 527 y Ft(\()p Fr(r)s Ft(\))30 b(in)g Fk(D)690 494 y Fn(0)687 556 y Fp(\027)726 537 y Fd(0)753 527 y Ft(,)h Fr(q)d Fk(\025)d Ft(0,)31 b(suc)m(h)f(that,)h (if)f(w)m(e)h(put)692 756 y Fr(H)775 717 y Fn(\000)768 785 y Fp(k)805 794 y Fl(1)839 785 y Fp(;\022)892 794 y Fl(1)930 756 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)1247 669 y Fm(X)1251 865 y Fp(q)r Fn(\025)p Fq(0)1394 756 y Fr(H)1477 717 y Fn(\000)1470 785 y Fp(k)1507 794 y Fl(1)1541 785 y Fp(;\022)1594 794 y Fl(1)1628 785 y Fp(;q)1685 756 y Ft(\()p Fr(r)s Ft(\))p Fr(z)1845 718 y Fp(q)1945 756 y Fr(;)76 b(H)2129 717 y Fq(+)2122 785 y Fp(k)2159 794 y Fl(1)2192 785 y Fp(;\022)2245 794 y Fl(1)2284 756 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)2601 669 y Fm(X)2604 865 y Fp(q)r Fn(\025)p Fq(0)2747 756 y Fr(H)2830 717 y Fq(+)2823 785 y Fp(k)2860 794 y Fl(1)2894 785 y Fp(;\022)2947 794 y Fl(1)2981 785 y Fp(;q)3039 756 y Ft(\()p Fr(r)s Ft(\))p Fr(z)3199 718 y Fp(q)75 1041 y Ft(w)m(e)k(ha)m(v)m(e)75 1154 y Fo(a\))f Ft(F)-8 b(or)31 b Fr(\022)c Fk(2)e Ft(\()p Fr(\022)597 1116 y Fn(\000)594 1180 y Fq(1)656 1154 y Fr(;)15 b(\022)739 1168 y Fq(1)778 1154 y Ft(\),)31 b(the)g(series)767 1366 y Fk(j)p Fr(H)875 1327 y Fn(\000)868 1395 y Fp(k)905 1404 y Fl(1)939 1395 y Fp(;\022)1018 1366 y Fk(\000)20 b Fr(H)1192 1327 y Fn(\000)1185 1395 y Fp(k)1222 1404 y Fl(1)1256 1395 y Fp(;\022)1309 1404 y Fl(1)1347 1366 y Fk(j)1372 1328 y Fp(\027)1411 1305 y Fd(0)1434 1328 y Fp(;d)1494 1366 y Ft(\()p Fr(R)1598 1380 y Fq(2)1638 1366 y Ft(\))26 b(=)1794 1279 y Fm(X)1798 1475 y Fp(q)r Fn(\025)p Fq(0)1941 1366 y Fk(jj)p Fr(H)2074 1327 y Fn(\000)2067 1395 y Fp(k)2104 1404 y Fl(1)2138 1395 y Fp(;\022)r(;q)2251 1366 y Ft(\()p Fr(r)s Ft(\))20 b Fk(\000)g Fr(H)2559 1327 y Fn(\000)2552 1395 y Fp(k)2589 1404 y Fl(1)2623 1395 y Fp(;\022)2676 1404 y Fl(1)2710 1395 y Fp(;q)2768 1366 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2932 1381 y Fp(\027)2971 1362 y Fd(0)2994 1381 y Fp(;d)3054 1366 y Fr(R)3124 1321 y Fp(q)3123 1392 y Fq(2)75 1651 y Ft(satis\014es)1345 1764 y(lim)1233 1825 y Fp(\022)r Fn(!)p Fp(\022)1372 1834 y Fl(1)1406 1825 y Fp(;\022)r(<\022)1549 1834 y Fl(1)1598 1764 y Fk(j)p Fr(H)1706 1726 y Fn(\000)1699 1793 y Fp(k)1736 1802 y Fl(1)1770 1793 y Fp(;\022)1849 1764 y Fk(\000)g Fr(H)2023 1726 y Fn(\000)2016 1793 y Fp(k)2053 1802 y Fl(1)2087 1793 y Fp(;\022)2140 1802 y Fl(1)2178 1764 y Fk(j)2203 1726 y Fp(\027)2242 1703 y Fd(0)2265 1726 y Fp(;d)2325 1764 y Ft(\()p Fr(R)2429 1778 y Fq(2)2469 1764 y Ft(\))26 b(=)e(0)p Fr(:)75 1983 y Fo(b\))30 b Ft(F)-8 b(or)31 b Fr(\022)d Fk(2)c Ft(\()p Fr(\022)601 1997 y Fq(1)641 1983 y Fr(;)15 b(\022)727 1945 y Fq(+)724 2009 y(1)786 1983 y Ft(\),)31 b(the)f(series)767 2195 y Fk(j)p Fr(H)875 2157 y Fq(+)868 2224 y Fp(k)905 2233 y Fl(1)939 2224 y Fp(;\022)1018 2195 y Fk(\000)20 b Fr(H)1192 2157 y Fq(+)1185 2224 y Fp(k)1222 2233 y Fl(1)1256 2224 y Fp(;\022)1309 2233 y Fl(1)1347 2195 y Fk(j)1372 2158 y Fp(\027)1411 2134 y Fd(0)1434 2158 y Fp(;d)1494 2195 y Ft(\()p Fr(R)1598 2209 y Fq(2)1638 2195 y Ft(\))26 b(=)1794 2109 y Fm(X)1798 2305 y Fp(q)r Fn(\025)p Fq(0)1941 2195 y Fk(jj)p Fr(H)2074 2157 y Fq(+)2067 2224 y Fp(k)2104 2233 y Fl(1)2138 2224 y Fp(;\022)r(;q)2251 2195 y Ft(\()p Fr(r)s Ft(\))20 b Fk(\000)g Fr(H)2559 2157 y Fq(+)2552 2224 y Fp(k)2589 2233 y Fl(1)2623 2224 y Fp(;\022)2676 2233 y Fl(1)2710 2224 y Fp(;q)2768 2195 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2932 2210 y Fp(\027)2971 2191 y Fd(0)2994 2210 y Fp(;d)3054 2195 y Fr(R)3124 2151 y Fp(q)3123 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Fq(\))3794 2809 y Fk([)96 2921 y Ft(\026)75 2944 y Fr(D)150 2958 y Fq(1)190 2944 y Ft(,)43 b(so)e(that)h(the)f(function)f Fr(t)i Fk(7!)h Fo(h)1388 2959 y Fp(k)1425 2968 y Fl(1)1464 2944 y Ft(\()p Fr(t)1532 2911 y Fq(2)1571 2944 y Ft(\))e(is)g(holomorphic)g (and)f(b)s(ounded)f(on)3001 2921 y(\026)2985 2944 y Fr(S)3041 2975 y Fp(\022)3074 2984 y Fl(1)3109 2975 y Fp(;)p Fq(\()p Fp(\022)3191 2945 y Fl(+)3189 2996 y(1)3241 2975 y Fn(\000)p Fp(\022)3329 2984 y Fl(1)3364 2975 y Fq(\))3422 2944 y Fk([)3531 2921 y Ft(\026)3510 2944 y Fr(D)3585 2958 y Fq(1)3625 2944 y Ft(.)72 b(F)-8 b(or)75 3083 y(ev)m(ery)30 b(\014xed)f Fr(\022)e Fk(2)e Ft(\()p Fr(\022)770 3045 y Fn(\000)767 3109 y Fq(1)829 3083 y Fr(;)15 b(\022)915 3045 y Fq(+)912 3109 y(1)973 3083 y Ft(\),)31 b(w)m(e)f(consider)f(the) g(function)h Fr(h)2113 3098 y Fp(k)2150 3107 y Fl(1)2184 3098 y Fp(;\022)2243 3083 y Ft(\()p Fr(r)s Ft(\))25 b(=)g Fo(h)2536 3098 y Fp(k)2573 3107 y Fl(1)2612 3083 y Ft(\()p Fr(r)2691 3050 y Fq(2)2730 3083 y Fr(e)2772 3050 y Fq(2)p Fp(i\022)2871 3083 y Ft(\))30 b(whic)m(h)f(is)g(in)g(particular)h(a)75 3196 y(b)s(ounded)e Fr(C)518 3163 y Fn(1)592 3196 y Fk(\000)p Ft(function)i(on)g Fi(R)1211 3210 y Fq(+)1300 3196 y Ft(with)h(v)-5 b(alues)30 b(in)g Fi(C)p Ft(.)41 b(W)-8 b(e)31 b(mak)m(e)h(the)e(follo)m(wing)i(assumptions.)75 3309 y Fo(i\))h Ft(There)h(exist)g Fr(\026;)15 b(\024)32 b(>)e Ft(1)p Fr(=\027)1084 3276 y Fn(0)1142 3309 y Ft(suc)m(h)j(that)i (for)e(all)i(0)d Fk(\024)e Fr(k)2048 3323 y Fq(1)2119 3309 y Fk(\024)h Fr(S)c Fk(\000)22 b Ft(1,)35 b Fr(h)2554 3324 y Fp(k)2591 3333 y Fl(1)2626 3324 y Fp(;\022)2685 3309 y Ft(\()p Fr(r)s Ft(\))f(are)g Fr(C)3060 3276 y Fn(1)3134 3309 y Fk(\000)p Ft(functions)f(on)h Fi(R)3796 3323 y Fq(+)75 3422 y Ft(suc)m(h)c(that)1372 3555 y Fk(j)p Fr(h)1449 3507 y Fq(\()p Fp(q)r Fq(\))1449 3584 y Fp(k)1486 3593 y Fl(1)1521 3584 y Fp(;\022)1580 3555 y Ft(\()p Fr(r)s Ft(\))p Fk(j)25 b(\024)g Fr(C)1905 3570 y Fp(k)1942 3579 y Fl(1)2222 3493 y Fr(q)s Ft(!)p 1991 3534 532 4 v 1991 3617 a(\()p Fr(\024)p Ft(\()p Fr(r)f Ft(+)c Fr(\026)p Ft(\)\))2394 3591 y Fq(1+)p Fp(q)2532 3555 y Fr(;)75 3776 y Ft(for)30 b(a)h Fr(C)355 3791 y Fp(k)392 3800 y Fl(1)456 3776 y Fr(>)25 b Ft(0,)31 b(for)f(all)h Fr(q)d Fk(\025)d Ft(0,)31 b(all)g Fr(r)d Fk(2)d Fi(R)1531 3790 y Fq(+)1620 3776 y Ft(and)30 b(all)h Fr(\022)c Fk(2)e Ft(\()p Fr(\022)2160 3738 y Fn(\000)2157 3802 y Fq(1)2219 3776 y Fr(;)15 b(\022)2305 3738 y Fq(+)2302 3802 y(1)2364 3776 y Ft(\).)75 3889 y Fo(ii\))30 b Ft(F)-8 b(or)31 b(all)g Fr(\022)d Fk(2)d Ft(\()p Fr(\022)731 3851 y Fn(\000)728 3915 y Fq(1)789 3889 y Fr(;)15 b(\022)875 3851 y Fq(+)872 3915 y(1)934 3889 y Ft(\)\))31 b(and)f(0)25 b Fk(\024)g Fr(k)1425 3903 y Fq(1)1490 3889 y Fk(\024)g Fr(S)h Fk(\000)19 b Ft(1,)32 b(one)e(can)h(c)m(ho)s(ose)g(a)g(constan)m(t)h Fr(C)2989 3904 y Fp(k)3026 3913 y Fl(1)3060 3904 y Fp(;\022)3113 3913 y Fl(1)3147 3904 y Fp(;\022)3231 3889 y Fr(>)25 b Ft(0)31 b(suc)m(h)f(that)1081 4132 y Fk(j)p Fr(h)1158 4084 y Fq(\()p Fp(q)r Fq(\))1158 4161 y Fp(k)1195 4170 y Fl(1)1230 4161 y Fp(;\022)1283 4170 y Fl(1)1321 4132 y Ft(\()p Fr(r)s Ft(\))21 b Fk(\000)f Fr(h)1599 4084 y Fq(\()p Fp(q)r Fq(\))1599 4161 y Fp(k)1636 4170 y Fl(1)1670 4161 y Fp(;\022)1729 4132 y Ft(\()p Fr(r)s Ft(\))p Fk(j)26 b(\024)f Fr(C)2055 4147 y Fp(k)2092 4156 y Fl(1)2126 4147 y Fp(;\022)2179 4156 y Fl(1)2213 4147 y Fp(;\022)2513 4070 y Fr(q)s Ft(!)p 2282 4111 V 2282 4194 a(\()p Fr(\024)p Ft(\()p Fr(r)f Ft(+)c Fr(\026)p Ft(\)\))2685 4168 y Fq(1+)p Fp(q)2823 4132 y Fr(;)75 4371 y Ft(for)30 b(all)h Fr(q)e Fk(\025)24 b Ft(0)31 b(and)f(all)h Fr(r)d Fk(2)d Fi(R)1105 4385 y Fq(+)1164 4371 y Ft(,)30 b(with)g(the)h(additional)g(prop)s(ert) m(y)f(that)1657 4566 y(lim)1634 4627 y Fp(\022)r Fn(!)p Fp(\022)1773 4636 y Fl(1)1822 4566 y Fr(C)1887 4581 y Fp(k)1924 4590 y Fl(1)1958 4581 y Fp(;\022)2011 4590 y Fl(1)2045 4581 y Fp(;\022)2129 4566 y Ft(=)25 b(0)p Fr(:)75 4852 y Fc(7.2)112 b(Hyp)s(erfunctions)39 b(solutions)f(for)f 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1242 y Fr(V)2380 1204 y Fq(+)2360 1271 y Fp(k)r(;\022)2457 1242 y Ft(\()p Fr(r)s Ft(\))p Fr(z)2617 1209 y Fp(k)2660 1242 y Fr(=k)s Ft(!.)42 b(As)31 b(a)f(result,)h(the)g(functions)75 1371 y Fr(r)d Fk(7!)d Fr(V)333 1333 y Fq(+)313 1400 y Fp(k)r(;\022)410 1371 y Ft(\()p Fr(r)s Ft(\))31 b(are)f(con)m(tin)m(uous)i(functions)d(from)h ([0)p Fr(;)15 b Ft(+)p Fk(1)p Ft(\))32 b(in)m(to)f Fi(C)p Ft(.)40 b(Moreo)m(v)m(er,)33 b(w)m(e)e(ha)m(v)m(e)1490 1582 y(sup)1432 1665 y Fp(r)r Fn(2)p Fq([0)p Fp(;)p Fn(1)p Fq(\))1700 1582 y Fk(j)p Fr(V)1799 1544 y Fq(+)1778 1612 y Fp(k)r(;\022)1876 1582 y Ft(\()p Fr(r)s Ft(\))p Fk(j)p Fr(e)2057 1545 y Fn(\000)p Fp(\027)t(r)2215 1582 y Fr(<)25 b Ft(+)p Fk(1)p Fr(:)75 1860 y Ft(In)31 b(particular,)h(from)f(the)h (prop)s(osition)f(2,)i(the)f(functions)f Fr(r)f Fk(7!)d Fr(V)2405 1822 y Fn(\000)2385 1889 y Fp(k)r(;\022)2482 1860 y Ft(\()p Fr(r)s Ft(\))32 b(and)f Fr(r)f Fk(7!)d Fr(V)3068 1822 y Fq(+)3048 1889 y Fp(k)r(;\022)3145 1860 y Ft(\()p Fr(r)s Ft(\),)32 b Fr(k)f Fk(\025)c Ft(0,)32 b(b)s(elong)75 1989 y(to)g(the)f(Banac)m(h)h(space)g Fk(D)986 1956 y Fn(0)983 2017 y Fp(\027)1022 1998 y Fd(0)1079 1989 y Ft(for)f(an)m(y)h Fr(\027)1443 1956 y Fn(0)1492 1989 y Fr(>)26 b(\027)6 b Ft(,)31 b(and)g Fr(V)1947 1950 y Fn(\000)1927 2018 y Fp(\022)2006 1989 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))33 b(and)d Fr(V)2483 1950 y Fq(+)2463 2018 y Fp(\022)2542 1989 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))33 b(are)f(con)m(v)m(ergen)m(t)h(series)f(in)e(the)75 2102 y(v)-5 b(ariable)31 b Fr(z)k Ft(with)30 b(v)-5 b(alues)31 b(in)f Fk(D)1150 2069 y Fn(0)1147 2130 y Fp(\027)1186 2111 y Fd(0)1243 2102 y Ft(for)g(an)m(y)g Fr(\027)1604 2069 y Fn(0)1653 2102 y Fr(>)25 b(\027)6 b Ft(.)216 2215 y(In)41 b(the)g(follo)m(wing)i(prop)s(osition,)g(w)m(e)f(study)e (tangen)m(tial)k(limit)e(prop)s(erties)f(in)f Fk(D)3119 2182 y Fn(0)3116 2243 y Fp(\027)3155 2224 y Fd(0)3223 2215 y Ft(of)i(the)f(solutions)75 2327 y Fo(W)184 2294 y Fn(\000)244 2327 y Ft(\()p Fr(t)312 2294 y Fq(2)351 2327 y Fr(;)15 b(z)t Ft(\))32 b(and)d Fo(W)789 2294 y Fq(+)849 2327 y Ft(\()p Fr(t)917 2294 y Fq(2)956 2327 y Fr(;)15 b(z)t Ft(\).)75 2531 y Fo(Prop)s(osition)36 b(11)46 b Fh(L)-5 b(et)33 b(the)g(assumptions)i Fo(\(h1'\))p Fh(,)d Fo(\(h2'\))p Fh(,)g Fo(\(h3'\))g Fh(hold.)44 b(Then,)33 b(ther)-5 b(e)33 b(exist)g(series)878 2785 y Fr(V)951 2747 y Fn(\000)931 2815 y Fp(\022)964 2824 y Fl(1)1010 2785 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)1327 2699 y Fm(X)1328 2896 y Fp(k)r Fn(\025)p Fq(0)1474 2785 y Fr(V)1547 2747 y Fn(\000)1527 2815 y Fp(k)r(;\022)1619 2824 y Fl(1)1656 2785 y Ft(\()p Fr(r)s Ft(\))1780 2724 y Fr(z)1826 2691 y Fp(k)p 1780 2764 89 4 v 1787 2848 a Fr(k)s Ft(!)1945 2785 y Fr(;)80 b(V)2123 2747 y Fq(+)2103 2815 y Fp(\022)2136 2824 y Fl(1)2182 2785 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)2499 2699 y Fm(X)2501 2896 y Fp(k)r Fn(\025)p Fq(0)2646 2785 y Fr(V)2719 2747 y Fq(+)2699 2815 y Fp(k)r(;\022)2791 2824 y Fl(1)2829 2785 y Ft(\()p Fr(r)s Ft(\))2953 2724 y Fr(z)2999 2691 y Fp(k)p 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y(its)k(solution)g(whic)m(h)f(is)g(con)m(v)m (ergen)m(t)j(in)d(the)h(v)-5 b(ariable)31 b Fr(z)e Fk(2)c Fr(D)2188 1497 y Fp(R)2241 1506 y Fl(2)2280 1483 y Ft(,)31 b Fr(R)2405 1497 y Fq(2)2470 1483 y Fr(>)25 b Ft(0)30 b(small)h(enough,)g(with)f(co)s(e\016cien)m(ts)75 1595 y(in)d Fk(D)251 1563 y Fn(0)248 1624 y Fp(\027)287 1605 y Fd(0)313 1595 y Ft(.)40 b(Lik)m(ewise,)29 b(the)e(Cauc)m(h)m(y)h (problem)e(\(50\))j(and)e(\(51\))h(in)f(the)g(space)h Fk(D)2712 1563 y Fn(0)2709 1624 y Fp(\027)2748 1605 y Fd(0)2802 1595 y Ft(also)g(mak)m(e)g(sense)f(for)g Fr(\022)g Ft(=)e Fr(\022)3790 1609 y Fq(1)3829 1595 y Ft(.)75 1708 y(Let)1464 1857 y Fr(V)1537 1819 y Fq(+)1517 1887 y Fp(\022)1550 1896 y Fl(1)1596 1857 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))27 b(=)1913 1771 y Fm(X)1915 1969 y Fp(k)r Fn(\025)p Fq(0)2060 1857 y Fr(V)2133 1819 y Fq(+)2113 1887 y Fp(k)r(;\022)2205 1896 y Fl(1)2243 1857 y Ft(\()p Fr(r)s Ft(\))2367 1796 y Fr(z)2413 1763 y Fp(k)p 2367 1836 V 2374 1920 a Fr(k)s Ft(!)75 2114 y(its)k(solution)g(whic)m(h)f(is)g(con)m(v)m(ergen)m(t)j (in)d(the)h(v)-5 b(ariable)31 b Fr(z)e Fk(2)c Fr(D)2188 2128 y Fp(R)2241 2137 y Fl(2)2280 2114 y Ft(,)31 b Fr(R)2405 2128 y Fq(2)2470 2114 y Fr(>)25 b Ft(0)30 b(small)h(enough,)g(with)f (co)s(e\016cien)m(ts)75 2227 y(in)g Fk(D)254 2194 y Fn(0)251 2255 y Fp(\027)290 2236 y Fd(0)317 2227 y Ft(.)216 2340 y Fh(In)j(the)g(fol)5 b(lowing,)33 b(we)g(wil)5 b(l)33 b(show)h(that)g(ther)-5 b(e)33 b(exists)g Fr(R)2159 2354 y Fq(3)2224 2340 y Fr(>)25 b Ft(0)33 b Fh(smal)5 b(l)34 b(enough)f(such)f(that)75 2453 y Fo(a\))1224 2647 y Ft(lim)1112 2708 y Fp(\022)r Fn(!)p Fp(\022)1251 2717 y Fl(1)1285 2708 y Fp(;\022)r(<\022)1428 2717 y Fl(1)1480 2533 y Fq(+)p Fn(1)1477 2561 y Fm(X)1478 2758 y Fp(k)r Fq(=0)1623 2647 y Fk(jj)p Fr(V)1747 2609 y Fn(\000)1726 2676 y Fp(k)r(;\022)1824 2647 y Ft(\()p Fr(r)s Ft(\))21 b Fk(\000)f Fr(V)2123 2609 y Fn(\000)2103 2676 y Fp(k)r(;\022)2195 2685 y Fl(1)2232 2647 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2396 2662 y 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y(and)h(will)g(not)h(b)s(e)f (detailed.)216 3676 y(F)-8 b(or)33 b(ev)m(ery)f(\014xed)f Fr(\022)f Fk(2)d Ft(\()p Fr(\022)1085 3638 y Fn(\000)1082 3702 y Fq(1)1143 3676 y Fr(;)15 b(\022)1226 3690 y Fq(1)1266 3676 y Ft(\),)32 b(w)m(e)g(denote)h(b)m(y)e Fr(V)1988 3638 y Fn(\000)1968 3705 y Fp(\022)2001 3714 y Fl(1)2036 3705 y Fp(;\022)2094 3676 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))30 b(=)d Fr(V)2489 3638 y Fn(\000)2469 3705 y Fp(\022)2502 3714 y Fl(1)2548 3676 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))23 b Fk(\000)e Fr(V)2930 3638 y Fn(\000)2910 3705 y Fp(\022)2989 3676 y Ft(\()p Fr(r)m(;)15 b(z)t Ft(\))33 b(and)e(w)m(e)i(will)f(also) 75 3789 y(use)e(the)h(notations)184 3993 y Fr(h)236 4008 y Fp(k)273 4017 y Fl(1)308 4008 y Fp(;\022)361 4017 y Fl(1)395 4008 y Fp(;\022)453 3993 y Ft(\()p Fr(r)s Ft(\))26 b(=)f Fr(e)731 3956 y Fq(2)p Fp(i\022)823 3965 y Fl(1)862 3993 y Fr(h)914 4008 y Fp(k)951 4017 y Fl(1)986 4008 y Fp(;\022)1039 4017 y Fl(1)1077 3993 y Ft(\()p Fr(r)s Ft(\))c Fk(\000)f Fr(e)1345 3956 y 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y Ft(+)20 b Fk(j)p Fr(d)2036 4534 y Fp(\022)r(;\022)2124 4543 y Fl(1)2163 4519 y Fk(j)2188 4482 y Fp(\027)2227 4458 y Fd(0)2250 4482 y Fp(;d)2310 4519 y Ft(\()p Fr(R)2414 4533 y Fq(2)2453 4519 y Ft(\))26 b(=)f(0)p Fr(;)75 4770 y Fh(for)33 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fh(.)41 b Ft(First)30 b(of)h(all,)g(the)g(assumption)f Fo(\(h1'\))g Ft(sa)m(ys)h(that)75 4988 y(\(55\))1443 b(lim)1567 5049 y Fp(\022)r Fn(!)p Fp(\022)1706 5058 y Fl(1)1740 5049 y Fp(;\022)r(<\022)1883 5058 y Fl(1)1932 4988 y Fr(w)1999 4944 y Fp(\027)2038 4921 y Fd(0)2061 4944 y Fp(;d)1997 5018 y(j;\022)2079 5027 y Fl(1)2112 5018 y Fp(;\022)2196 4988 y Ft(=)25 b(0)p Fr(:)75 5247 y Ft(Moreo)m(v)m(er,)33 b(for)d(all)h Fr(q)d Fk(\025)d Ft(0,)31 b(w)m(e)g(write)g Fr(d)1445 5262 y Fp(q)r(;\022)r(;\022)1587 5271 y Fl(1)1625 5247 y Ft(\()p Fr(r)s Ft(\))25 b(=)g Fr(d)1907 5214 y Fq(1)1907 5275 y Fp(q)r(;\022)r(;\022)2049 5284 y Fl(1)2087 5247 y Ft(\()p Fr(r)s Ft(\))20 b(+)g Fr(d)2359 5214 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Ft(\()p Fr(r)s Ft(\))p 2732 5534 451 4 v 2905 5618 a Fr(q)2946 5632 y Fq(2)2985 5618 y Ft(!)3193 5555 y(\))p eop end %%Page: 22 22 TeXDict begin 22 21 bop 3764 100 a Ft(22)75 399 y(and)368 664 y Fr(d)415 627 y Fq(2)415 687 y Fp(q)r(;\022)r(;\022)557 696 y Fl(1)595 664 y Ft(\()p Fr(r)s Ft(\))26 b(=)843 551 y Fp(S)t Fn(\000)p Fq(1)846 578 y Fm(X)831 775 y Fp(k)868 784 y Fl(1)902 775 y Fq(=0)1080 578 y Fm(X)1007 769 y Fp(q)1039 778 y Fl(1)1073 769 y Fq(+)p Fp(q)1160 778 y Fl(2)1194 769 y Fq(=)p Fp(q)1298 664 y Fr(h)1350 679 y Fp(k)1387 688 y Fl(1)1422 679 y Fp(;\022)1481 664 y Ft(\()p Fr(r)s Ft(\))p Fr(e)1637 627 y Fq(2)p Fp(i\022)1736 664 y Fr(@)1789 627 y Fn(\000)p Fq(1)1784 687 y Fp(r)1883 664 y Ft(\(\()p Fr(H)2036 626 y Fn(\000)2029 694 y Fp(k)2066 703 y Fl(1)2101 694 y Fp(;\022)2154 703 y Fl(1)2188 694 y Fp(;q)2240 703 y Fl(1)2278 664 y Ft(\()p Fr(r)s Ft(\))20 b Fk(\000)g Fr(H)2586 626 y Fn(\000)2579 694 y Fp(k)2616 703 y Fl(1)2650 694 y Fp(;\022)r(;q)2757 703 y Fl(1)2794 664 y 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Fp(q)r(;\022)r(;\022)496 1926 y Fl(1)534 1894 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)698 1909 y Fp(\027)737 1891 y Fd(0)760 1909 y Fp(;d)846 1894 y Fk(\024)954 1781 y Fp(S)t Fn(\000)p Fq(1)957 1808 y Fm(X)941 2005 y Fp(k)978 2014 y Fl(1)1013 2005 y Fq(=0)1190 1808 y Fm(X)1118 1999 y Fp(q)1150 2008 y Fl(1)1184 1999 y Fq(+)p Fp(q)1271 2008 y Fl(2)1305 1999 y Fq(=)p Fp(q)1409 1894 y Fr(C)1474 1909 y Fp(\026;\024;\027)1636 1891 y Fd(0)1662 1894 y Fr(C)1727 1909 y Fp(k)1764 1918 y Fl(1)1802 1894 y Fk(jj)p Fr(H)1935 1856 y Fn(\000)1928 1924 y Fp(k)1965 1933 y Fl(1)2000 1924 y Fp(;\022)2053 1933 y Fl(1)2087 1924 y Fp(;q)2139 1933 y Fl(1)2177 1894 y Ft(\()p Fr(r)s Ft(\))g Fk(\000)g Fr(H)2485 1856 y Fn(\000)2478 1924 y Fp(k)2515 1933 y Fl(1)2549 1924 y Fp(;\022)r(;q)2656 1933 y Fl(1)2693 1894 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)2857 1909 y Fp(\027)2896 1891 y Fd(0)2920 1909 y Fp(;d)2990 1816 y Fk(jj)p Fr(V)3113 1778 y Fn(\000)3093 1846 y Fp(q)3125 1855 y Fl(2)3159 1846 y Fq(+)p Fp(k)3251 1855 y Fl(1)3285 1846 y Fp(;\022)3338 1855 y Fl(1)3377 1816 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)3541 1832 y Fp(\027)3580 1813 y Fd(0)3603 1832 y Fp(;d)p 2990 1873 V 3274 1957 a Fr(q)3315 1971 y Fq(2)3354 1957 y Ft(!)75 2184 y(for)30 b(all)h Fr(q)e Fk(\025)24 b Ft(0.)42 b(W)-8 b(e)31 b(deduce)f(that)75 2420 y(\(56\))85 b Fk(j)p Fr(d)392 2383 y Fq(1)392 2443 y Fp(\022)r(;\022)480 2452 y Fl(1)518 2420 y Fk(j)543 2383 y Fp(\027)582 2359 y Fd(0)605 2383 y Fp(;d)665 2420 y Ft(\()p Fr(R)769 2434 y Fq(2)809 2420 y Ft(\))25 b(=)965 2334 y Fm(X)969 2530 y Fp(q)r Fn(\025)p Fq(0)1112 2420 y Fk(jj)p Fr(d)1209 2383 y Fq(1)1209 2443 y Fp(q)r(;\022)r(;\022)1351 2452 y Fl(1)1389 2420 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)1553 2435 y Fp(\027)1592 2416 y Fd(0)1616 2435 y Fp(;d)1676 2420 y Ft(\()p Fr(R)1780 2434 y Fq(2)1819 2420 y Ft(\))1854 2383 y Fp(q)1918 2420 y Fk(\024)1246 2639 y Fp(S)t Fn(\000)p Fq(1)1249 2667 y Fm(X)1234 2864 y Fp(k)1271 2873 y Fl(1)1305 2864 y Fq(=0)1410 2753 y Fr(C)1475 2768 y Fp(\026;\024;\027)1637 2749 y Fd(0)1663 2753 y Ft(\()p Fk(j)p Fr(e)1765 2715 y Fp(i\022)1822 2724 y Fl(1)1882 2753 y Fk(\000)20 b Fr(e)2015 2715 y Fp(i\022)2078 2753 y Fk(j)p Fr(C)2168 2768 y Fp(k)2205 2777 y Fl(1)2264 2753 y Ft(+)g Fr(C)2420 2768 y Fp(k)2457 2777 y Fl(1)2491 2768 y Fp(;\022)2544 2777 y Fl(1)2578 2768 y Fp(;\022)2637 2753 y Ft(\))p Fk(j)p Fr(H)7 b Fk(j)2805 2709 y Fp(\027)2844 2685 y Fd(0)2867 2709 y Fp(;d)2805 2782 y(k)2842 2791 y Fl(1)2877 2782 y Fp(;\022)2930 2791 y Fl(1)2968 2753 y Ft(\()p Fr(R)3072 2767 y Fq(2)3112 2753 y Ft(\)\()p Fr(@)3235 2715 y Fp(k)3272 2724 y Fl(1)3230 2775 y Fp(z)3312 2753 y Fk(j)p Fr(V)20 b Fk(j)3435 2709 y Fp(\027)3474 2685 y Fd(0)3497 2709 y Fp(;d)3435 2782 y(\022)3468 2791 y Fl(1)3557 2753 y Ft(\)\()p Fr(R)3696 2767 y Fq(2)3736 2753 y Ft(\))75 3043 y(and)75 3309 y(\(57\))89 b Fk(j)p Fr(d)396 3271 y Fq(2)396 3332 y Fp(\022)r(;\022)484 3341 y Fl(1)523 3309 y Fk(j)548 3271 y Fp(\027)587 3248 y Fd(0)609 3271 y Fp(;d)669 3309 y Ft(\()p Fr(R)773 3323 y Fq(2)813 3309 y Ft(\))26 b(=)970 3223 y Fm(X)973 3419 y Fp(q)r Fn(\025)p Fq(0)1116 3309 y Fk(jj)p Fr(d)1213 3271 y Fq(2)1213 3332 y Fp(q)r(;\022)r(;\022)1355 3341 y Fl(1)1394 3309 y Ft(\()p Fr(r)s Ft(\))p Fk(jj)1558 3324 y Fp(\027)1597 3305 y Fd(0)1620 3324 y Fp(;d)1680 3309 y Ft(\()p Fr(R)1784 3323 y Fq(2)1824 3309 y Ft(\))1859 3271 y Fp(q)1922 3309 y Fk(\024)2031 3195 y Fp(S)t Fn(\000)p Fq(1)2033 3223 y Fm(X)2018 3420 y Fp(k)2055 3429 y Fl(1)2090 3420 y Fq(=0)2195 3309 y Fr(C)2260 3324 y Fp(\026;\024;\027)2422 3305 y Fd(0)2448 3309 y Fr(C)2513 3324 y Fp(k)2550 3333 y Fl(1)2588 3309 y Fk(j)p Fr(H)7 b Fk(j)2721 3265 y Fp(\027)2760 3241 y Fd(0)2783 3265 y Fp(;d)2721 3338 y(k)2758 3347 y Fl(1)2793 3338 y Fp(;\022)r(;\022)2901 3347 y Fl(1)2939 3309 y Ft(\()p Fr(R)3043 3323 y Fq(2)3083 3309 y Ft(\)\()p Fr(@)3206 3271 y Fp(k)3243 3280 y Fl(1)3201 3331 y Fp(z)3282 3309 y Fk(j)p Fr(V)21 b Fk(j)3406 3265 y Fp(\027)3445 3241 y Fd(0)3467 3265 y Fp(;d)3406 3338 y(\022)3439 3347 y Fl(1)3527 3309 y Ft(\)\()p Fr(R)3666 3323 y Fq(2)3707 3309 y Ft(\))p Fr(:)75 3606 y Ft(F)-8 b(rom)31 b(the)f(assumption)g Fo(\(h2'\))p Ft(,)h(w)m(e)f(ha)m(v)m(e)1533 3822 y(lim)1421 3883 y Fp(\022)r Fn(!)p Fp(\022)1560 3892 y Fl(1)1594 3883 y Fp(;\022)r(<\022)1737 3892 y Fl(1)1786 3822 y Fk(j)p Fr(H)7 b Fk(j)1919 3778 y Fp(\027)1958 3754 y Fd(0)1981 3778 y Fp(;d)1919 3851 y(k)1956 3860 y Fl(1)1991 3851 y Fp(;\022)r(;\022)2099 3860 y Fl(1)2137 3822 y Ft(\()p Fr(R)2241 3836 y Fq(2)2281 3822 y Ft(\))25 b(=)g(0)p Fr(:)75 4071 y Ft(F)-8 b(rom)31 b(the)f(assumption)g Fo(\(h3'\))p Ft(,)h(w)m(e)f(ha)m(v)m(e)1657 4273 y(lim)1545 4334 y Fp(\022)r Fn(!)p Fp(\022)1684 4343 y Fl(1)1718 4334 y Fp(;\022)r(<\022)1861 4343 y Fl(1)1910 4273 y Fr(C)1975 4288 y Fp(k)2012 4297 y Fl(1)2047 4288 y Fp(;\022)2100 4297 y Fl(1)2134 4288 y Fp(;\022)2218 4273 y Ft(=)25 b(0)p Fr(:)75 4522 y Ft(So)30 b(that,)h(from)f(\(56\))i(and)e(\(57\),)i (w)m(e)f(deduce)f(that)1218 4724 y(lim)1106 4786 y Fp(\022)r Fn(!)p Fp(\022)1245 4795 y Fl(1)1279 4786 y Fp(;\022)r(<\022)1422 4795 y Fl(1)1471 4724 y Fk(j)p Fr(d)1543 4687 y Fq(1)1543 4748 y Fp(\022)r(;\022)1631 4757 y Fl(1)1670 4724 y Fk(j)1695 4687 y Fp(\027)1734 4663 y Fd(0)1757 4687 y Fp(;d)1817 4724 y Ft(\()p Fr(R)1921 4738 y Fq(2)1960 4724 y Ft(\))21 b(+)f Fk(j)p Fr(d)2179 4687 y Fq(2)2179 4748 y Fp(\022)r(;\022)2267 4757 y Fl(1)2306 4724 y Fk(j)2331 4687 y Fp(\027)2370 4663 y Fd(0)2392 4687 y Fp(;d)2452 4724 y Ft(\()p Fr(R)2556 4738 y Fq(2)2596 4724 y Ft(\))26 b(=)f(0)p Fr(:)75 4989 y Ft(F)-8 b(rom)31 b(the)f(inequalit)m(y)i Fk(j)p Fr(d)964 5004 y Fp(\022)r(;\022)1052 5013 y Fl(1)1091 4989 y Fk(j)1116 4956 y Fp(\027)1155 4933 y Fd(0)1178 4956 y Fp(;d)1238 4989 y Ft(\()p Fr(R)1342 5003 y Fq(2)1381 4989 y Ft(\))26 b Fk(\024)f(j)p Fr(d)1610 4956 y Fq(1)1610 5017 y Fp(\022)r(;\022)1698 5026 y Fl(1)1737 4989 y Fk(j)1762 4956 y Fp(\027)1801 4933 y Fd(0)1823 4956 y Fp(;d)1883 4989 y Ft(\()p Fr(R)1987 5003 y Fq(2)2027 4989 y Ft(\))c(+)f Fk(j)p Fr(d)2246 4956 y Fq(2)2246 5017 y Fp(\022)r(;\022)2334 5026 y Fl(1)2372 4989 y Fk(j)2397 4956 y Fp(\027)2436 4933 y Fd(0)2459 4956 y Fp(;d)2519 4989 y Ft(\()p Fr(R)2623 5003 y Fq(2)2663 4989 y Ft(\),)31 b(w)m(e)g(deduce)f(that)75 5216 y(\(58\))1313 b(lim)1437 5278 y Fp(\022)r Fn(!)p Fp(\022)1576 5287 y Fl(1)1610 5278 y Fp(;\022)r(<\022)1753 5287 y Fl(1)1802 5216 y Fk(j)p Fr(d)1874 5231 y Fp(\022)r(;\022)1962 5240 y Fl(1)2000 5216 y Fk(j)2025 5179 y Fp(\027)2064 5155 y Fd(0)2087 5179 y Fp(;d)2147 5216 y Ft(\()p Fr(R)2251 5230 y Fq(2)2291 5216 y Ft(\))25 b(=)g(0)75 5465 y(Finally)-8 b(,)30 b(from)e(\(55\))h(and)f(\(58\),)i(w)m(e)e(get)h(the)g(result)f (\(54\))h(and)f(the)g(limit)h Fo(a\))e Ft(of)h(prop)s(osition)g(11)h (follo)m(ws.)41 b Fg(2)216 5627 y Ft(In)36 b(the)h(next)g(theorem,)i(w) m(e)f(get)g(h)m(yp)s(erfunctions)d(solutions)i(for)g(the)g(Cauc)m(h)m (y)g(problems)f(\(44\),)41 b(\(45\),)75 5740 y(\(46\),)32 b(\(47\).)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 3764 100 a Ft(23)75 399 y Fo(Theorem)35 b(1)46 b 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Fo(Theorem)35 b(2)46 b Fh(We)33 b(c)-5 b(onsider)35 b(the)g(fol)5 b(lowing)35 b(two)g(inte)-5 b(gr)g(o-di\013er)g(ential)36 b(Cauchy)e(pr)-5 b(oblems)36 b(with)f(irr)-5 b(e)g(gular)75 1773 y(singularity)34 b(at)f Fr(t)25 b Ft(=)g(0)p Fh(,)75 2045 y Ft(\(66\))241 b(4)p Fr(t)554 2008 y Fq(2)593 2045 y Fr(@)641 2059 y Fp(t)671 2045 y Fr(@)724 2008 y Fp(S)719 2068 y(z)775 2045 y Fo(X)854 2008 y Fn(\000)914 2045 y Ft(\()p Fr(t;)15 b(z)t Ft(\))21 b(+)f(\(2)p Fr(t)h Ft(+)f(2\))p Fr(@)1573 2008 y Fp(S)1568 2068 y(z)1625 2045 y Fo(X)1704 2008 y Fn(\000)1763 2045 y Ft(\()p Fr(t;)15 b(z)t Ft(\))26 b(=)2086 1932 y Fp(S)t Fn(\000)p Fq(1)2089 1959 y Fm(X)2074 2157 y Fp(k)2111 2166 y Fl(1)2145 2157 y Fq(=0)2251 1922 y Fm(Z)2342 1948 y Fp(t)2301 2128 y Fq(0)2386 2045 y Fo(A)2465 2007 y Fn(\000)2465 2075 y Fp(k)2502 2084 y Fl(1)2541 2045 y Ft(\()p Fr(t)20 b Fk(\000)g Fr(\034)5 b(;)15 b(z)t Ft(\))p Fr(@)2939 2008 y Fp(k)2976 2017 y Fl(1)2934 2068 y Fp(z)3016 2045 y Fo(X)3095 2008 y Fn(\000)3154 2045 y 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2842 4536 a Fr(k)s Ft(!)2935 4474 y Fr(:)75 4801 y Fo(Pro)s(of)31 b Ft(Let)111 5050 y Fo(W)220 5012 y Fn(\000)219 5077 y Fp(j)279 5050 y Ft(\()p Fr(t)p Ft(\))26 b(=)f Fo(W)613 5012 y Fq(+)612 5077 y Fp(j)673 5050 y Ft(\()p Fr(t)p Ft(\))h(=)e Fo(W)1005 5064 y Fp(j)1042 5050 y Ft(\()p Fr(t)p Ft(\))i(=)1270 4937 y Fq(+)p Fn(1)1267 4964 y Fm(X)1268 5162 y Fp(k)r Fq(=0)1436 4987 y Fo(w)1512 5002 y Fp(j;k)p 1423 5029 192 4 v 1423 5113 a Ft(\(2)p Fr(k)s Ft(\)!)1625 5050 y Fr(t)1658 5013 y Fp(k)1761 5050 y Fr(;)76 b Fo(H)1944 5012 y Fn(\000)1944 5080 y Fp(k)1981 5089 y Fl(1)2019 5050 y Ft(\()p Fr(t;)15 b(z)t Ft(\))27 b(=)e Fo(H)2413 5012 y Fq(+)2413 5080 y Fp(k)2450 5089 y Fl(1)2488 5050 y Ft(\()p Fr(t;)15 b(z)t Ft(\))26 b(=)f Fo(H)2881 5065 y Fp(k)2918 5074 y Fl(1)2957 5050 y Ft(\()p Fr(t;)15 b(z)t Ft(\))26 b(=)3271 4937 y Fq(+)p Fn(1)3268 4964 y Fm(X)3269 5162 y Fp(k)r Fq(=0)3425 4987 y Fo(h)3483 5002 y Fp(k)3520 5011 y Fl(1)3554 5002 y Fp(;k)3616 4987 y Ft(\()p Fr(z)t Ft(\))p 3425 5029 309 4 v 3483 5113 a(\(2)p Fr(k)s Ft(\)!)3743 5050 y Fr(t)3776 5013 y Fp(k)75 5336 y Ft(b)s(e)34 b(the)h(T)-8 b(a)m(ylor)35 b(expansions)g(of)f Fo(W)1332 5298 y Fn(\000)1331 5363 y Fp(j)1392 5336 y Ft(\()p Fr(t)p Ft(\),)i Fo(W)1665 5298 y Fq(+)1664 5363 y Fp(j)1725 5336 y Ft(\()p Fr(t)p Ft(\))f(and)f Fo(H)2126 5298 y Fn(\000)2126 5365 y Fp(k)2163 5374 y Fl(1)2201 5336 y Ft(\()p Fr(t;)15 b(z)t Ft(\),)37 b Fo(H)2534 5298 y Fq(+)2534 5365 y Fp(k)2571 5374 y Fl(1)2610 5336 y Ft(\()p Fr(t;)15 b(z)t Ft(\))35 b(for)g Fk(j)p Fr(t)p Fk(j)d Fr(<)g Ft(1.)54 b(W)-8 b(e)36 b(in)m(tro)s(duce)75 5449 y(the)31 b(formal)f(series)1006 5597 y(^)989 5620 y Fo(X)1068 5634 y Fp(j)1104 5620 y Ft(\()p Fr(t)p Ft(\))c(=)1332 5506 y Fq(+)p Fn(1)1329 5534 y Fm(X)1330 5731 y Fp(k)r Fq(=0)1486 5557 y Fo(w)1562 5572 y Fp(j;k)p 1486 5599 167 4 v 1531 5682 a Fr(k)s Ft(!)1662 5620 y Fr(t)1695 5582 y Fp(k)1798 5620 y Fr(;)1916 5597 y Ft(^)1899 5620 y Fo(A)1978 5635 y Fp(k)2015 5644 y Fl(1)2053 5620 y Ft(\()p Fr(t;)15 b(z)t Ft(\))27 b(=)2368 5506 y Fq(+)p Fn(1)2365 5534 y Fm(X)2366 5731 y Fp(k)r Fq(=0)2521 5557 y Fo(h)2579 5572 y Fp(k)2616 5581 y Fl(1)2651 5572 y Fp(;k)2713 5557 y Ft(\()p Fr(z)t Ft(\))p 2521 5599 309 4 v 2638 5682 a Fr(k)s Ft(!)2840 5620 y Fr(t)2873 5582 y Fp(k)2915 5620 y Fr(:)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 3764 100 a Ft(27)75 399 y Fo(Lemma)35 b(8)45 b Fh(The)33 b(formal)i(series)1535 531 y Ft(^)1518 554 y Fo(X)p Ft(\()p Fr(t;)15 b(z)t Ft(\))27 b(=)1909 468 y Fm(X)1910 665 y Fp(k)r Fn(\025)p Fq(0)2065 493 y Fo(w)2141 508 y Fp(k)2183 493 y Ft(\()p Fr(z)t Ft(\))p 2065 533 236 4 v 2145 617 a Fr(k)s Ft(!)2310 554 y Fr(t)2343 517 y Fp(k)2386 554 y Fr(;)75 819 y Fh(wher)-5 b(e)34 b Fo(w)408 834 y Fp(k)451 819 y Ft(\()p Fr(z)t Ft(\))p Fh(,)g Fr(k)c Fk(\025)c Ft(0)p Fh(,)34 b(ar)-5 b(e)34 b(holomorphic)j(functions)c(on)h(a)g(disc)g Fr(D)2451 833 y Fp(R)2504 842 y Fl(1)2576 819 y Fh(c)-5 b(ontinuous)34 b(on)3187 796 y Ft(\026)3165 819 y Fr(D)3240 833 y Fp(R)3293 842 y Fl(1)3332 819 y Fh(,)g(satis\014es)g(the)75 932 y(formal)g(Cauchy)f(pr)-5 b(oblem)75 1210 y Ft(\(70\))329 b(4)p Fr(t)642 1172 y Fq(2)682 1210 y Fr(@)730 1224 y Fp(t)760 1210 y Fr(@)813 1172 y Fp(S)808 1232 y(z)881 1187 y Ft(^)864 1210 y Fo(X)p Ft(\()p Fr(t;)15 b(z)t Ft(\))21 b(+)f(\(2)p Fr(t)h Ft(+)f(2\))p Fr(@)1602 1172 y Fp(S)1597 1232 y(z)1671 1187 y Ft(^)1654 1210 y Fo(X)p Ft(\()p Fr(t;)15 b(z)t Ft(\))27 b(=)2057 1096 y Fp(S)t Fn(\000)p Fq(1)2060 1124 y Fm(X)2044 1321 y Fp(k)2081 1330 y Fl(1)2116 1321 y Fq(=0)2221 1086 y Fm(Z)2312 1113 y Fp(t)2272 1292 y Fq(0)2374 1187 y Ft(^)2357 1210 y Fo(A)2436 1225 y Fp(k)2473 1234 y Fl(1)2511 1210 y Ft(\()p Fr(t)21 b Fk(\000)f Fr(\034)5 b(;)15 b(z)t Ft(\))p Fr(@)2910 1172 y Fp(k)2947 1181 y Fl(1)2905 1232 y Fp(z)3003 1187 y Ft(^)2986 1210 y Fo(X)q Ft(\()p Fr(\034)5 b(;)15 b(z)t Ft(\))p Fr(d\034)75 1495 y Fh(with)33 b(initial)h(c)-5 b(onditions)75 1692 y Ft(\(71\))973 b(\()p Fr(@)1296 1654 y Fp(j)1291 1715 y(z)1350 1669 y Ft(^)1333 1692 y Fo(X)p Ft(\)\()p Fr(t;)15 b Ft(0\))27 b(=)1775 1669 y(^)1758 1692 y Fo(X)1837 1706 y Fp(j)1874 1692 y Ft(\()p Fr(t)p Ft(\))66 b Fr(;)80 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 1897 y Fh(if)27 b(and)h(only)g(if)e(its)i(asso)-5 b(ciate)g(d)29 b(mo)-5 b(di\014e)g(d)30 b(Bor)-5 b(el)28 b(tr)-5 b(ansform)29 b(of)f(or)-5 b(der)29 b(1,)f Ft(\()p Fr(t;)15 b(z)t Ft(\))26 b Fk(7!)g Fo(W)q Ft(\()p Fr(t)3093 1864 y Fq(2)3133 1897 y Fr(;)15 b(z)t Ft(\))26 b(=)f(\()p Fk(B)3471 1911 y Fq(1)3527 1874 y Ft(^)3510 1897 y Fo(X)q Ft(\)\()p Fr(t)3693 1864 y Fq(2)3732 1897 y Fr(;)15 b(z)t Ft(\))75 2010 y Fh(satis\014es)34 b(the)f(Cauchy)g(pr)-5 b(oblem)75 2288 y Ft(\(72\))368 b Fr(@)656 2251 y Fp(S)651 2311 y(z)707 2288 y Fo(W)q Ft(\()p Fr(t)884 2251 y Fq(2)924 2288 y Fr(;)15 b(z)t Ft(\))26 b(=)1180 2175 y Fp(S)t Fn(\000)p Fq(1)1182 2202 y Fm(X)1167 2400 y Fp(k)1204 2409 y Fl(1)1239 2400 y Fq(=0)1446 2227 y Ft(1)p 1354 2267 229 4 v 1354 2351 a Fr(t)1387 2324 y Fq(2)1446 2351 y Ft(+)20 b(2)1608 2165 y Fm(Z)1699 2191 y Fp(t)1658 2371 y Fq(0)1744 2165 y Fm(Z)1834 2191 y Fp(t)1859 2200 y Fl(1)1794 2371 y Fq(0)1914 2288 y Fo(H)1996 2303 y Fp(k)2033 2312 y Fl(1)2071 2288 y Ft(\(\()p Fr(t)2174 2302 y Fq(1)2234 2288 y Fk(\000)g Fr(t)2358 2302 y Fq(2)2397 2288 y Ft(\))2432 2251 y Fq(2)2472 2288 y Fr(;)15 b(z)t Ft(\))p Fr(@)2646 2251 y Fp(k)2683 2260 y Fl(1)2641 2311 y Fp(z)2723 2288 y Fo(W)q Ft(\()p Fr(t)2900 2251 y Fq(2)2900 2311 y(2)2940 2288 y Fr(;)g(z)t Ft(\))p Fr(dt)3141 2302 y Fq(2)3182 2288 y Fr(dt)3262 2302 y Fq(1)3301 2288 y Fr(;)75 2573 y Fh(with)33 b(initial)h(c)-5 b(onditions)75 2770 y Ft(\(73\))560 b(\()p Fr(@)883 2733 y Fp(j)878 2793 y(z)920 2770 y Fo(W)q Ft(\)\()p Fr(t)1132 2733 y Fq(2)1172 2770 y Fr(;)15 b Ft(0\))27 b(=)e Fo(W)1523 2784 y Fp(j)1559 2770 y Ft(\()p Fr(t)1627 2733 y Fq(2)1667 2770 y Ft(\))h(=)f(\()p Fr(@)1912 2733 y Fp(j)1907 2793 y(z)1949 2770 y Fk(B)2009 2784 y Fq(1)2065 2747 y Ft(^)2048 2770 y Fo(X)p Ft(\)\()p Fr(t)2230 2733 y Fq(2)2270 2770 y Fr(;)15 b Ft(0\))66 b Fr(;)81 b Ft(0)25 b Fk(\024)g Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 2967 y Fh(for)33 b(al)5 b(l)33 b Ft(\()p Fr(t;)15 b(z)t Ft(\))27 b Fk(2)e Fr(D)727 2981 y Fq(1)787 2967 y Fk(\002)19 b Fr(D)952 2981 y Fp(R)1005 2990 y Fl(1)1045 2967 y Fh(.)75 3196 y Fo(Pro)s(of)31 b Ft(F)-8 b(rom)31 b(the)g(classical)h(iden)m (tit)m(y)1025 3321 y Fm(Z)1116 3347 y Fp(t)1076 3527 y Fq(0)1146 3445 y Ft(\()p Fr(t)20 b Fk(\000)g Fr(\034)10 b Ft(\))1410 3407 y Fp(m)1472 3416 y Fl(1)1511 3445 y Fr(\034)1561 3407 y Fp(m)1623 3416 y Fl(2)1663 3445 y Fr(d\034)35 b Ft(=)2048 3383 y Fr(m)2128 3397 y Fq(1)2167 3383 y Ft(!)p Fr(m)2272 3397 y Fq(2)2311 3383 y Ft(!)p 1891 3424 603 4 v 1891 3507 a(\(1)21 b(+)f Fr(m)2163 3521 y Fq(1)2222 3507 y Ft(+)g Fr(m)2393 3521 y Fq(2)2433 3507 y Ft(\)!)2503 3445 y Fr(t)2536 3407 y Fp(m)2598 3416 y Fl(1)2633 3407 y Fq(+)p Fp(m)2750 3416 y Fl(2)2785 3407 y Fq(+1)2879 3445 y Fr(;)75 3685 y Ft(for)30 b Fr(m)294 3699 y Fq(1)333 3685 y Fr(;)15 b(m)453 3699 y Fq(2)518 3685 y Fk(\025)25 b Ft(0,)31 b(w)m(e)g(get)755 3812 y Fm(Z)846 3838 y Fp(t)806 4018 y Fq(0)908 3913 y Ft(^)891 3936 y Fo(A)970 3951 y Fp(k)1007 3960 y Fl(1)1046 3936 y Ft(\()p Fr(t)20 b Fk(\000)g Fr(\034)5 b(;)15 b(z)t Ft(\))1409 3913 y(^)1391 3936 y Fo(X)q Ft(\()p Fr(\034)5 b(;)15 b(z)t Ft(\))p Fr(d\034)37 b Ft(=)1892 3849 y Fm(X)1893 4047 y Fp(k)r Fn(\025)p Fq(1)2023 3936 y Ft(\()2178 3849 y Fm(X)2058 4047 y Fp(q)2090 4056 y Fl(1)2125 4047 y Fq(+)p Fp(q)2212 4056 y Fl(2)2246 4047 y Fq(=)p Fp(k)r Fn(\000)p Fq(1)2444 3936 y Fo(h)2502 3951 y Fp(k)2539 3960 y Fl(1)2574 3951 y Fp(;q)2626 3960 y Fl(1)2664 3936 y Ft(\()p Fr(z)t Ft(\))p Fo(w)2856 3950 y Fp(q)2888 3959 y Fl(2)2926 3936 y Ft(\()p Fr(z)t Ft(\)\))3087 3874 y Fr(t)3120 3841 y Fp(k)p 3087 3915 76 4 v 3087 3998 a Fr(k)s Ft(!)75 4241 y(and)30 b(also)h(if)f Fo(W)q Ft(\()p Fr(t;)15 b(z)t Ft(\))27 b(=)938 4172 y Fm(P)1034 4267 y Fp(k)r Fn(\025)p Fq(0)1182 4241 y Fo(w)1258 4256 y Fp(k)1301 4241 y Ft(\()p Fr(z)t Ft(\))p Fr(t)1450 4208 y Fp(k)1493 4241 y Fr(=)p Ft(\(2)p Fr(k)s Ft(\)!,)424 4388 y Fm(Z)514 4414 y Fp(t)474 4594 y Fq(0)559 4388 y Fm(Z)650 4414 y Fp(t)675 4423 y Fl(1)610 4594 y Fq(0)730 4512 y Fo(H)812 4527 y Fp(k)849 4536 y Fl(1)887 4512 y Ft(\(\()p Fr(t)990 4526 y Fq(1)1050 4512 y Fk(\000)20 b Fr(\034)10 b Ft(\))1226 4474 y Fq(2)1266 4512 y Fr(;)15 b(z)t Ft(\))p Fo(W)q Ft(\()p Fr(\034)1581 4474 y Fq(2)1622 4512 y Fr(;)g(z)t Ft(\))p Fr(d\034)10 b(dt)1920 4526 y Fq(1)1986 4512 y Ft(=)2082 4425 y Fm(X)2084 4623 y Fp(k)r Fn(\025)p Fq(1)2214 4512 y Ft(\()2369 4425 y Fm(X)2249 4623 y Fp(q)2281 4632 y Fl(1)2315 4623 y Fq(+)p Fp(q)2402 4632 y Fl(2)2436 4623 y Fq(=)p Fp(k)r Fn(\000)p Fq(1)2635 4512 y Fo(h)2693 4527 y Fp(k)2730 4536 y Fl(1)2764 4527 y Fp(;q)2816 4536 y Fl(1)2854 4512 y Ft(\()p Fr(z)t Ft(\))p Fo(w)3046 4526 y Fp(q)3078 4535 y Fl(2)3117 4512 y Ft(\()p Fr(z)t Ft(\)\))3320 4450 y Fr(t)3353 4417 y Fq(2)p Fp(k)p 3278 4491 192 4 v 3278 4574 a Ft(\(2)p Fr(k)s Ft(\)!)3481 4512 y Fr(:)75 4800 y Ft(On)30 b(the)g(other)h(hand,)1021 5051 y(\(4)p Fr(t)1134 5013 y Fq(2)1174 5051 y Fr(@)1222 5065 y Fp(t)1272 5051 y Ft(+)20 b(2)p Fr(t)p Ft(\))1494 5028 y(^)1476 5051 y Fo(X)q Ft(\()p Fr(t;)15 b(z)t Ft(\))26 b(=)1867 4965 y Fm(X)1869 5162 y Fp(k)r Fn(\025)p Fq(0)1999 5051 y Ft(\(2)p Fr(k)s Ft(\)\(2)p Fr(k)f Fk(\000)20 b Ft(1\))p Fo(w)2563 5066 y Fp(k)r Fn(\000)p Fq(1)2696 5051 y Ft(\()p Fr(z)t Ft(\))2822 4989 y Fr(t)2855 4956 y Fp(k)p 2822 5030 76 4 v 2822 5113 a Fr(k)s Ft(!)75 5338 y(and)1107 5487 y Fr(t)1140 5449 y Fq(2)1179 5487 y Fo(W)q Ft(\()p Fr(t)1356 5449 y Fq(2)1397 5487 y Fr(;)15 b(z)t Ft(\))26 b(=)1640 5401 y Fm(X)1641 5598 y Fp(k)r Fn(\025)p Fq(0)1771 5487 y Ft(\(2)p Fr(k)s Ft(\)\(2)p Fr(k)g Fk(\000)20 b Ft(1\))p Fo(w)2336 5502 y Fp(k)r Fn(\000)p Fq(1)2469 5487 y Ft(\()p Fr(z)t Ft(\))2636 5425 y Fr(t)2669 5392 y Fq(2)p Fp(k)p 2595 5466 192 4 v 2595 5549 a Ft(\(2)p Fr(k)s Ft(\)!)2797 5487 y Fr(;)75 5740 y Ft(whic)m(h)30 b(yields)h(the)f(result.)2761 b Fg(2)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 3764 100 a Ft(28)216 399 y(F)-8 b(rom)31 b(the)g(prop)s(osition)f(10,)h(w)m(e)g(see)g(that)g(the)f (Cauc)m(h)m(y)h(problem)75 672 y(\(74\))309 b Fr(@)597 634 y Fp(S)592 694 y(z)648 672 y Fo(W)757 634 y Fn(\000)816 672 y Ft(\()p Fr(t)884 634 y Fq(2)924 672 y Fr(;)15 b(z)t Ft(\))26 b(=)1180 558 y Fp(S)t Fn(\000)p Fq(1)1182 585 y Fm(X)1167 783 y Fp(k)1204 792 y Fl(1)1239 783 y Fq(=0)1446 610 y Ft(1)p 1354 651 229 4 v 1354 734 a Fr(t)1387 708 y Fq(2)1446 734 y Ft(+)20 b(2)1608 548 y Fm(Z)1699 574 y Fp(t)1658 754 y Fq(0)1744 548 y Fm(Z)1834 574 y Fp(t)1859 583 y Fl(1)1794 754 y Fq(0)1914 672 y Fo(H)1996 633 y Fn(\000)1996 701 y Fp(k)2033 710 y Fl(1)2071 672 y Ft(\(\()p Fr(t)2174 686 y Fq(1)2234 672 y Fk(\000)g Fr(t)2358 686 y Fq(2)2397 672 y Ft(\))2432 634 y Fq(2)2472 672 y Fr(;)15 b(z)t Ft(\))p Fr(@)2646 634 y Fp(k)2683 643 y Fl(1)2641 694 y Fp(z)2723 672 y Fo(W)2832 634 y Fn(\000)2892 672 y Ft(\()p Fr(t)2960 634 y Fq(2)2960 694 y(2)2999 672 y Fr(;)g(z)t Ft(\))p Fr(dt)3200 686 y Fq(2)3241 672 y Fr(dt)3321 686 y Fq(1)3361 672 y Fr(;)75 952 y Ft(with)30 b(initial)i(conditions)75 1143 y(\(75\))867 b(\()p Fr(@)1190 1106 y Fp(j)1185 1166 y(z)1227 1143 y Fo(W)1336 1106 y Fn(\000)1396 1143 y Ft(\)\()p Fr(t)1499 1106 y Fq(2)1539 1143 y Fr(;)15 b Ft(0\))26 b(=)f Fo(W)1890 1105 y Fn(\000)1889 1171 y Fp(j)1950 1143 y Ft(\()p Fr(t)2018 1106 y Fq(2)2057 1143 y Ft(\))61 b Fr(;)76 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 1350 y Ft(has)41 b(a)h(unique)e(solution)i(\()p Fr(t;)15 b(z)t Ft(\))44 b Fk(7!)1364 1282 y Fm(P)1460 1377 y Fp(k)r Fn(\025)p Fq(0)1608 1350 y Fo(W)1717 1317 y Fn(\000)1776 1350 y Ft(\()p Fr(t)1844 1317 y Fq(2)1884 1350 y Fr(;)15 b(z)t Ft(\))44 b(=)2164 1282 y Fm(P)2259 1377 y Fp(k)r Fn(\025)p Fq(0)2407 1350 y Fo(W)2516 1312 y Fn(\000)2515 1379 y Fp(k)2576 1350 y Ft(\()p Fr(t)2644 1317 y Fq(2)2684 1350 y Ft(\))p Fr(z)2765 1317 y Fp(k)2808 1350 y Fr(=k)s Ft(!)e(whic)m(h)f(is)g(a)h(con)m(v)m(ergen)m(t)75 1475 y(series)32 b(in)g Fr(z)g Fk(2)27 b Fr(D)665 1489 y Fp(R)718 1498 y Fl(1)789 1475 y Ft(with)k(co)s(e\016cien)m(ts)j Fr(t)28 b Fk(7!)f Fo(W)1747 1436 y Fn(\000)1746 1504 y Fp(k)1807 1475 y Ft(\()p Fr(t)1875 1442 y Fq(2)1914 1475 y Ft(\))33 b(b)s(elonging)f(to)g(the)g(spaces)g Fk(A)3018 1505 y Fp(\034)3057 1486 y Fd(\000)3109 1505 y Fp(;\016)3162 1475 y Fd(\000)3160 1515 y Fj(n)3215 1505 y Fp(;\027)3277 1475 y Ft(,)h(for)e(all)i Fr(n)27 b Fk(\025)h Ft(1.)75 1588 y(Lik)m(ewise,)k(from)e(the)g(prop)s(osition) g(10,)i(the)e(Cauc)m(h)m(y)h(problem)75 1861 y(\(76\))309 b Fr(@)597 1823 y Fp(S)592 1883 y(z)648 1861 y Fo(W)757 1823 y Fq(+)816 1861 y Ft(\()p Fr(t)884 1823 y Fq(2)924 1861 y Fr(;)15 b(z)t Ft(\))26 b(=)1180 1747 y Fp(S)t Fn(\000)p Fq(1)1182 1774 y Fm(X)1167 1972 y Fp(k)1204 1981 y Fl(1)1239 1972 y Fq(=0)1446 1799 y Ft(1)p 1354 1840 V 1354 1923 a Fr(t)1387 1897 y Fq(2)1446 1923 y Ft(+)20 b(2)1608 1737 y Fm(Z)1699 1763 y Fp(t)1658 1943 y Fq(0)1744 1737 y Fm(Z)1834 1763 y Fp(t)1859 1772 y Fl(1)1794 1943 y Fq(0)1914 1861 y Fo(H)1996 1822 y Fq(+)1996 1890 y Fp(k)2033 1899 y Fl(1)2071 1861 y Ft(\(\()p Fr(t)2174 1875 y Fq(1)2234 1861 y Fk(\000)g Fr(t)2358 1875 y Fq(2)2397 1861 y Ft(\))2432 1823 y Fq(2)2472 1861 y Fr(;)15 b(z)t Ft(\))p Fr(@)2646 1823 y Fp(k)2683 1832 y Fl(1)2641 1883 y Fp(z)2723 1861 y Fo(W)2832 1823 y Fq(+)2892 1861 y Ft(\()p Fr(t)2960 1823 y Fq(2)2960 1883 y(2)2999 1861 y Fr(;)g(z)t Ft(\))p Fr(dt)3200 1875 y Fq(2)3241 1861 y Fr(dt)3321 1875 y Fq(1)3361 1861 y Fr(;)75 2141 y Ft(with)30 b(initial)i(conditions)75 2332 y(\(77\))867 b(\()p Fr(@)1190 2295 y Fp(j)1185 2355 y(z)1227 2332 y Fo(W)1336 2295 y Fq(+)1396 2332 y Ft(\)\()p Fr(t)1499 2295 y Fq(2)1539 2332 y Fr(;)15 b Ft(0\))26 b(=)f Fo(W)1890 2294 y Fq(+)1889 2360 y Fp(j)1950 2332 y Ft(\()p Fr(t)2018 2295 y Fq(2)2057 2332 y Ft(\))61 b Fr(;)76 b Ft(0)26 b Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(S)g Fk(\000)20 b Ft(1)p Fr(:)75 2539 y Ft(has)41 b(a)h(unique)e(solution)i(\()p Fr(t;)15 b(z)t Ft(\))44 b Fk(7!)1364 2471 y Fm(P)1460 2566 y Fp(k)r Fn(\025)p Fq(0)1608 2539 y Fo(W)1717 2506 y Fq(+)1776 2539 y Ft(\()p Fr(t)1844 2506 y Fq(2)1884 2539 y Fr(;)15 b(z)t Ft(\))44 b(=)2164 2471 y Fm(P)2259 2566 y Fp(k)r Fn(\025)p Fq(0)2407 2539 y Fo(W)2516 2501 y Fq(+)2515 2568 y Fp(k)2576 2539 y Ft(\()p Fr(t)2644 2506 y Fq(2)2684 2539 y Ft(\))p Fr(z)2765 2506 y Fp(k)2808 2539 y Fr(=k)s Ft(!)e(whic)m(h)f(is)g(a)h(con)m(v)m(ergen)m(t)75 2664 y(series)31 b(in)f Fr(z)f Fk(2)c Fr(D)657 2678 y Fp(R)710 2687 y Fl(1)780 2664 y Ft(with)30 b(co)s(e\016cien)m(ts)i Fr(t)25 b Fk(7!)g Fo(W)1730 2625 y Fn(\000)1729 2693 y Fp(k)1790 2664 y Ft(\()p Fr(t)1858 2631 y Fq(2)1897 2664 y Ft(\))31 b(b)s(elonging)f(to)i(the)e(spaces)h Fk(A)2994 2694 y Fp(\034)3033 2675 y Fd(\000)3085 2694 y Fp(;\016)3138 2664 y Fd(\000)3136 2704 y Fj(n)3190 2694 y Fp(;\027)3253 2664 y Ft(,)f(for)h(all)g Fr(n)25 b Fk(\025)g Ft(1.)216 2777 y(W)-8 b(e)38 b(deduce)f(that)h(the)f (condition)h Fo(i\))e Ft(of)h(Prop)s(osition)h(12)f(is)h(satis\014ed)f (for)g(the)g(function)f Fo(W)q Ft(\()p Fr(u;)15 b(z)t Ft(\))39 b(=)75 2889 y Fo(W)184 2856 y Fn(\000)244 2889 y Ft(\()p Fr(u;)15 b(z)t Ft(\))32 b(=)f Fo(W)695 2856 y Fq(+)754 2889 y Ft(\()p Fr(u;)15 b(z)t Ft(\))35 b(for)f(\()p Fr(u;)15 b(z)t Ft(\))32 b Fk(2)f Fr(D)1547 2903 y Fq(1)1609 2889 y Fk(\002)22 b Fr(D)1777 2903 y Fp(R)1830 2912 y Fl(1)1869 2889 y Ft(.)51 b(Moreo)m(v)m(er,)37 b(the)d(condition)h Fo(ii\))e Ft(of)h(Prop)s(osition)g(12)g(is)75 3002 y(satis\014ed)e(for) f(the)h(analytic)i(con)m(tin)m(uation)f(\()p Fr(u;)15 b(z)t Ft(\))29 b Fk(7!)e Fo(W)2062 2969 y Fn(\000)2122 3002 y Ft(\()p Fr(u;)15 b(z)t Ft(\))33 b(\(resp.)44 b(\()p Fr(u;)15 b(z)t Ft(\))29 b Fk(7!)f Fo(W)3095 2969 y Fq(+)3154 3002 y Ft(\()p Fr(u;)15 b(z)t Ft(\)\))34 b(of)e Fo(W)q Ft(\()p Fr(u;)15 b(z)t Ft(\))75 3115 y(with)30 b(resp)s(ect)g(to)i Fr(u)e Ft(in)g(the)h(direction)g Fr(\030)1473 3082 y Fn(\000)1557 3115 y Fk(2)25 b Ft(\(2)p Fr(\022)1769 3077 y Fn(\000)1766 3141 y Fq(1)1828 3115 y Fr(;)15 b Ft(2)p Fr(\022)1956 3129 y Fq(1)1996 3115 y Ft(\))30 b(\(resp.)41 b Fr(\030)2369 3082 y Fq(+)2453 3115 y Fk(2)25 b Ft(\(2)p Fr(\022)2662 3129 y Fq(1)2702 3115 y Fr(;)15 b Ft(2)p Fr(\022)2833 3077 y Fq(+)2830 3141 y(1)2892 3115 y Ft(\)\).)216 3228 y(So)30 b(that)g(the)g(formal)g(series)1239 3205 y(^)1223 3228 y Fo(X)p Ft(\()p Fr(t;)15 b(z)t Ft(\))26 b(=)1613 3160 y Fm(P)1709 3255 y Fp(k)r Fn(\025)p Fq(0)1857 3228 y Fo(w)1933 3243 y Fp(k)1975 3228 y Ft(\()p Fr(z)t Ft(\))p Fr(t)2124 3195 y Fp(k)2168 3228 y Fr(=k)s Ft(!)k(solution)h(of) f(the)g(formal)g(Cauc)m(h)m(y)g(problem)75 3341 y(\(70\),)47 b(\(71\),)h(is)42 b(1)p Fk(\000)p Ft(summable)g(in)g(directions)h Fr(\030)1790 3308 y Fn(\000)1892 3341 y Ft(and)e Fr(\030)2124 3308 y Fq(+)2226 3341 y Ft(as)h(formal)h(series)g(in)f(the)h(Banac)m(h) g(space)g(of)75 3454 y(holomorphic)36 b(functions)f(on)g Fr(D)1204 3468 y Fp(R)1257 3477 y Fl(2)1297 3454 y Ft(,)i 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Fo(H)157 4402 y Fn(\000)157 4470 y Fp(k)194 4479 y Fl(1)232 4440 y Ft(\()p Fr(u;)15 b(z)t Ft(\))35 b(=)d Fo(H)660 4402 y Fq(+)660 4470 y Fp(k)697 4479 y Fl(1)735 4440 y Ft(\()p Fr(u;)15 b(z)t Ft(\))37 b(for)d(\()p Fr(u;)15 b(z)t Ft(\))34 b Fk(2)f Fr(D)1534 4454 y Fq(1)1597 4440 y Fk(\002)23 b Fr(D)1766 4454 y Fp(R)1819 4463 y Fl(1)1858 4440 y Ft(.)54 b(Moreo)m(v)m(er,)39 b(the)c(condition)g Fo(ii\))g Ft(of)g(Prop)s(osition)g(12)g(is)75 4565 y(satis\014ed)30 b(for)g(the)h(analytic)g(con)m(tin)m(uation)h(\()p Fr(u;)15 b(z)t Ft(\))27 b Fk(7!)e Fo(H)2023 4527 y Fn(\000)2023 4595 y Fp(k)2060 4604 y Fl(1)2099 4565 y Ft(\()p Fr(u;)15 b(z)t Ft(\))31 b(\(resp.)40 b(\()p Fr(u;)15 b(z)t Ft(\))27 b Fk(7!)e Fo(H)3034 4527 y Fq(+)3034 4595 y Fp(k)3071 4604 y Fl(1)3110 4565 y Ft(\()p Fr(u;)15 b(z)t Ft(\)\))31 b(of)g Fo(H)3570 4580 y Fp(k)3607 4589 y Fl(1)3645 4565 y Ft(\()p Fr(u;)15 b(z)t Ft(\))75 4678 y(with)30 b(resp)s(ect)g(to)i Fr(u)e Ft(in)g(the)h(direction)g Fr(\030)1473 4645 y Fn(\000)1562 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Fl(1)1665 5017 y Ft(\()p Fr(t;)15 b(z)t Ft(\))37 b(the)f(1)p Fk(\000)p Ft(sum)f(in)h(the)g(direction)h Fr(\030)3070 4984 y Fn(\000)3165 5017 y Ft(and)f Fo(A)3427 4979 y Fq(+)3427 5046 y Fp(k)3464 5055 y Fl(1)3502 5017 y Ft(\()p Fr(t;)15 b(z)t Ft(\))37 b(the)75 5130 y(1)p Fk(\000)p Ft(sum)f(in)g(the)h(direction)g Fr(\030)1094 5097 y Fq(+)1153 5130 y Ft(.)60 b(F)-8 b(rom)37 b(the)g(remark)g(1,)h (the)f(1)p Fk(\000)p Ft(sums)f(can)h(b)s(e)f(expressed)g(as)h(an)g(in)m (tegral)75 5243 y(op)s(erator)31 b(in)m(v)m(olving)h(the)e(k)m(ernel)h (function)f Fr(z)g Fk(7!)25 b Fr(e)p Ft(\()p Fr(z)t Ft(\))31 b(as)g(in)f(\(64\))i(and)e(\(65\).)216 5356 y(F)-8 b(rom)46 b(the)f(general)i(summabilit)m(y)e(theory)-8 b(,)50 b(see)c([1)q(],)j (c)m(hapter)d(6.3,)k(w)m(e)c(deduce)f(that)h(the)f(1)p Fk(\000)p Ft(sums)75 5469 y Fo(X)154 5436 y Fn(\000)213 5469 y Ft(\()p Fr(t;)15 b(z)t Ft(\))32 b(satisfy)e(the)h(problem)f (\(66\),)i(\(67\))g(and)e Fo(X)1893 5436 y Fq(+)1952 5469 y Ft(\()p Fr(t;)15 b(z)t Ft(\))31 b(satisfy)g(the)g(problem)e (\(68\),)k(\(69\).)216 5627 y Fh(In)41 b(the)h(se)-5 b(c)g(ond)42 b(p)-5 b(art)43 b(of)f(the)f(pr)-5 b(o)g(of,)45 b(we)d(show)g(that)h(e)-5 b(ach)41 b Ft(1)p Fk(\000)p Fh(sums)h Fo(X)2755 5594 y Fn(\000)2814 5627 y Ft(\()p Fr(t;)15 b(z)t Ft(\))42 b Fh(and)g Fo(X)3309 5594 y Fq(+)3369 5627 y Ft(\()p Fr(t;)15 b(z)t Ft(\))42 b Fh(c)-5 b(an)42 b(b)-5 b(e)75 5740 y(expr)g(esse)g(d)34 b(as)g(gener)-5 b(alize)g(d)34 b(L)-5 b(aplac)g(e)34 b(tr)-5 b(ansform)35 b(of)e(stair)-5 b(c)g(ase)35 b(distributions)p Ft(.)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 3764 100 a Ft(29)216 399 y(F)-8 b(or)31 b(all)h Fr(\030)550 366 y Fn(\000)634 399 y Fk(2)25 b Ft(\(2)p Fr(\022)846 360 y Fn(\000)843 425 y Fq(1)905 399 y Fr(;)15 b Ft(2)p Fr(\022)1033 413 y Fq(1)1073 399 y Ft(\),)31 b(0)25 b Fr(<)g(\016)k(<)c(\031)s(=)p Ft(2,)32 b Fr(\032)25 b(>)g Ft(0,)31 b Fr(t)25 b Fk(2)g Fr(S)2166 418 y Fp(\030)2200 399 y Fd(\000)2251 418 y Fp(;\016)o(;\032)2361 399 y Ft(,)30 b Fr(z)g Fk(2)25 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b(\(79\))h(and)e(\(80\),)i(w)m(e)f(deduce)f(that) 1249 3684 y Fo(X)1328 3646 y Fn(\000)1387 3684 y Ft(\()p Fr(T)8 b(;)15 b(z)t Ft(\))26 b(=)1726 3597 y Fm(X)1728 3795 y Fp(k)r Fn(\025)p Fq(0)1873 3684 y Fk(L)1936 3699 y Fq(2)p Fp(\022)2004 3708 y Fl(1)2042 3684 y Ft(\()p Fr(V)2151 3645 y Fn(\000)2130 3713 y Fp(k)r(;\022)2222 3722 y Fl(1)2261 3684 y Ft(\()p Fr(r)s Ft(\)\)\()p Fr(T)13 b Ft(\))2556 3622 y Fr(z)2602 3589 y Fp(k)p 2556 3663 89 4 v 2563 3746 a Fr(k)s Ft(!)2655 3684 y Fr(;)75 3963 y Ft(for)30 b(all)h Fr(T)38 b Fk(2)25 b Fr(S)573 3978 y Fq(2)p Fp(\022)641 3987 y Fl(1)676 3978 y Fp(;\016)727 3990 y Fj(\022)756 4005 y Fl(1)795 3978 y Fp(;\032)855 3963 y Ft(,)30 b Fr(z)g Fk(2)25 b Fr(D)1143 3977 y Fp(R)1196 3986 y Fl(3)1235 3963 y Ft(.)40 b(Lik)m(ewise,)32 b(one)f(can)f(sho)m (w)h(with)f(similar)g(argumen)m(ts)h(that)1249 4229 y Fo(X)1328 4192 y Fq(+)1387 4229 y Ft(\()p Fr(T)8 b(;)15 b(z)t Ft(\))26 b(=)1726 4143 y Fm(X)1728 4341 y Fp(k)r Fn(\025)p Fq(0)1873 4229 y Fk(L)1936 4244 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(\023)-43 b(erisque,)32 b(59-60,)h(So)s(c.)d(Math.)h(F)-8 b(rance,)32 b(P)m(aris,)f(1978.)75 5035 y([15])47 b(W.)37 b(Rudin,)g Fh(F)-7 b(unctional)39 b(analysis.)f Ft(Second)e(edition.)i (In)m(ternational)g(Series)e(in)g(Pure)g(and)g(Applied)262 5148 y(Mathematics.)d(McGra)m(w-Hill,)g(Inc.,)d(New)h(Y)-8 b(ork,)31 b(1991.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0702270703186--