Équation homologique et classification analytique des germes de champs de vecteurs holomorphes de type noeud-col

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(1)Équation homologique et classification analytique des germes de champs de vecteurs holomorphes de type noeud-col Loïc Jean Dit Teyssier. To cite this version: Loïc Jean Dit Teyssier. Équation homologique et classification analytique des germes de champs de vecteurs holomorphes de type noeud-col. Mathématiques [math]. Université Rennes 1, 2003. Français. �tel-00005387�. HAL Id: tel-00005387 https://tel.archives-ouvertes.fr/tel-00005387 Submitted on 19 Mar 2004. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(26) GJI $# G &S$LÑS"% G &S$aS*$?OJ* %$*&. (0, 0). S2x=R ! SPÿQS*' I #$)x G N G G ! xS%$. Z(x, y) := A(x, y). ∂ ∂ + B(x, y) . ∂x ∂y. G 'v"r' I Sû$ R''R2xS%RòR I Þ2xSR E! $rm $LÑ&SO I N RQL ')""% I LÑ'[m N G #LMOJLÑ&Sut JG I $ $ I!S!hG $ G S& $r I N N N Z m G #LMOJLÑ&S×$ " %L 'H{==R½&S$aS%$# GfG # G &S&S"*%$ LÑ&S"RQLÑ*$RQS" I R½')*$ . f v r'aS. Z1 = λ 1 x. P& !HG $*)R. ∂ ∂ + λ2 y ∂x ∂y. λ2 6= 0 . λ :=. λ1 . λ2. GJI $;SLM G &S$ I × I EOJ* %$ÜS)2xSR E! $DSaÿJ*') I $ Z *' Z˜ $ G &('¼R½&=R N s'L~ I %&(' .zýTþ2ÿ } G N f N %&('21 G & v#I O I "%$ $ L N $ G &(' N R ¼"%#LÑ' I aR ! n%$ÜRÿ G LÑ·»hSr' I " I &%2x=RQ&OJ %&('ÜS GfG # G &S&S"*%$ N N .zýTþ2ÿ } G N 1gRQ&(' (0, 0) {Y *$#' f m f SLM# ψ G R Z˜ = ψ ∗ Z = Dψ −1 (Z ◦ ψ) .. q¤ÃOJ% S 9}k I L NMN r'RQOQû$LM&O I N LÑ%-LÑ&S I L ' ! RQ I &m') N 2x=R E! SÃÿQS*' I #$ Z $#%)R& G ')" FZ t q®%$ }k I L NMN *$aS]*]}k I L NÑN *')RQOJ$ G &(' N %$# GJI K %$×LÑ&(')"%OQ)R N %$aS Z {=SU$ G #'] I I &L2x=R E! SYÿQSr') I $ X LM&S I LÑ' N }k I L NMN *')RQOJP. $L{^*'$# I N %&('$#L {^L N -ÚLM$#' I & OJ% U S]} G &g*')L G &<x G N GG ! xS&S $S RQ&S& I N R½&' ! R½$)m N G #LMOJLÑ&S I &S I &SL ')"ix G N GG ! x>1Ü' N I Z = UX .. GJI $USLM G &S$' I vS I Þ2x=R E! $USEÿQS*' I #$ Z *' X $ G &'1RQ&=R N s')L~ I *&(' . ý2þ2ÿ G K L 'R N %&('v" I LÑÿ½R N %&(')$û$#L Z %$#'9 G & v#I O I "äm I &2x=R ! $ "%#LÑÿ½RQ&(' U X Guò U x G N GuEG ! xS .zýTþ2ÿ } G NÑN H1tf& ! RQ N %RUR N G $;SQ G & v#I O(R½LM$ G & . RQ&SR N s')L I GJI } G S I 5}k I L NÑN r'RQOJ*$St. } G MN N %&('-1 %$]' I &S I &SL ')" NÑN > 1 %&(' N %$. ! $S ÿJ*') I $©$ G &(' G & v#I O I "*$S{ q¤*$ GJIS! N %$ (λ1 , λ2) $ G &(' K LM*& $SLMS*&'L~ I *$ N G $n I ; I '2x=R E RQLÑ$ N LM&(ÿJ*$#;& %$]'©ÿfRQL³ I *&1OJ"*&S"%R N t GJI $S"%$#LM G &S$RQRu*'"%LÑ$* N *$ ! R%ÜS%$ G K LÑ')*$¤S%$2x=R ! $ S ÿJ*' I #$-S %$ λj $ GQI $ N Rur')L G &%$2x=R½&SOJ %&('$1S GÚG G &&S"%*$St ¤ *$ ! GJK N " R')L~ I *$ G &('1"*'"a G &S$#LMS"*"%*$UÿJ*$ N S" KSI '1 I $#L~ N *' G &(' ! G I L ' %$ ')Rÿ½R I ò')n%$ L !hG ]'RQ&('$S{Dm G *&%* ! RQr% I m S9Ü]t = G LM&gRQ#"û*'H{ *&(')ÃR I '*$S{Ü]t R I N Ru . & GJI $ #%&(ÿ G s G &S$+m N GJI ÿf)R½OJ t[ yS ±

(27) !HGJI I &û !hG $" E G S*&Sr'aS"*')RQL NÑN "_Si%*$')S2x&SL~ I *$21t=Å N $ G &(' ! R½Ú ! N EG &('"Q I N !hGJI λ ∈ C\R− L & s R I I &S GJI I G K LÑ'%$®$ GJI $ N Ru*'L G &%$SL»h" GG ! xSLM$ *$ N G R I Ät I RQ&S ! Z %$#' N LÑ&S"R½LM$R K N u{wu %$#' f m f SLM#aRQ&=R N s ')L~ I *&(' G & v#I O I "VmY$R ! R½#')LÑ ∗ N 2xSR E λ N LM&∈/"RQRLÑ− ∪ZNJ∪L N 1/N & s Ri I I & G K L ')ut I RQ&S. λ ∈ 1/N∗ Q{ L N *$#';RQ&=R N s ')L I %&(' G & v#I O I "#m I &%2x=R E! 1 I 'R½&' N %$·ÿ½RQ#L¥R K N %$ x S N R¢} G Z1 + εy1/λ ∂x∂ RÿQS ε ∈ {0, 1} N ¶%RQ$ λ ∈ N∗ %$#'Ü$#L L N R½LM *& ! % 1 N N ! N u{ *' y 1{P$ G ]')i I L &¸sÔR I P I G K L ')*$StYq G #$y I λ ∈ R− R1$L ' I R')L G &v& %$#' ! RQ$aR I $#$L®$L E - I & !hG LM&('S-ÿ I r} G N ®t ÆÚL λ *$#'YLM#)R½'L G &&S N { Z %$]'} G NÑN %&(' N LM&"RQ#LM$)R K N { RQLM$ N R ! "*$%&g%U ! *'LÑ')$LÑÿfLM$# I #$ ! I '? E ! S2x%V%*'#') / N LÑ&S= "R½LM$R½')L G &vS G &(ÿJ%#OJ%t ¤ %RQ$S{ N G LM&v*' G LÑ&S$¤"*'"¼RQ&=R N fs $" K L NMN R a %&(' ! RQ t "* f Á R½y G r' l t ¤ t G G s bSs b Yj

(28) "%$ G N I Q{ R_&S"RQ& E G ! xSLÑ$ R %$S (C, 0) {@m N R$ I LÑ'SY')Rÿ½R I & G & G LM&S$" N "*O(RQ&(')$_S ¤ t°ÆfLM%OQ N m. ')RÿJ%#$ N %$aSL»h" GuE . %& uµ¥ 1 R " N L G #"%$ ! RQÜÂØt t #v I & G SRQ&S$ N %$;RQ&S&S"*%$ ¡ ¦ !© i0²

(29) tgq G #$y I λ *$#'D)R½'L G &S& N { GJI K LÑ%& Z %$]' N LÑ&S"R½LM$R K N u{ GQI K LM*& L N %$#')} G NÑN *&' G K L 'R N %&('a"S I L ÿQR N %&('¢m I & *' I & $ I N.

(30) S%$ G g N *$S. = G LM&gRQ#R " f R I N Ru G & ! R½ N * )R 1&½ I f G N LM&"RQ#LM$)R K N *$ λ ∈ Q∗ 1-t RQ&S$ N i%RQ$a I &½ I f G N {B%*$ G − X0 := xk+1. jÕ. N . λ = 01a*'S1$# NÑN %$×"%$ G S& &=R½&'%$& G & g N *¶$ } G N $¼$ G &'_ G &S&S"%$ ! RQ. ∂ ∂ + y 1 + µxk ∂x ∂y. RÿQS (k, µ) ∈ N∗ tÛÜ]t R I N uR rê5Í!

(31) R«*&S$ I L ') ! GJI ÿQ"æ I I &E2x=R ! S 'Õs ! &½ I f G N * $#' RQ&=R N sf'L~ I * &'# G & v#I O I "'m I 5 & 2x=R E! S N RØ} G . . ¦gt¹¦gt 1. Z := U X X := X0 + xk+1 R U (0, 0) = λ2 ,. ∂ ∂y. ÿ #&%!')(*Yÿ*+!þ ,u&þ #.- .þ /!ýTÿ N G ! xSu{ N G LM& *' I &SL~ I t" þ2$ !þ Yþ*$#' I & OJ* ?Y*þ& ³ÿ S#gÿ } G &r'ý )L Gg&aÿ x G gGÿ rÿ./B?G-ÕþQÿ9;3- ÿ./E9>ÿA2ÿ$H.-ýTþIA&#ýJ?G-ëÿQÿ9;/³ÿ>#2þ.-@- þLK 9wýJ( þ = N I $#"S% a %&('S{ l t®Á R½#')LÑ&S*'*' l t f = t / R LM$ b ¨ u ¨ ©E

(32) *& £ ¥ G &('P G &S&S" I &rRQRu*'"%L f $)R½'L G &ÃS N R% N RQ$$#1RQ&SR N s')L . I U I & ' N }k I L NMN *'R½OJP%& '% Y LÑ&(ÿQR½L¥R½&'$V} G &*'L G &S& N $?RQ$#$ G %LM"*$Qm N x G N G & G LM¼ I k} I L NMN *'R½OJ ! RQ S%$DRQO Ig *&'$ S×&=R' I #V G x GG N G OJL~ I H1tJÅ N $ G &('D*&S$ I L ')_"*'%&S I %%$Ü')Rÿ½R I E R I %k} I N L NMN *')RQOJ%$"%$ G &S&=R½&'$D& G & N LM&S"%RQLÑ$)R K N %$ b ¨ )u ¨ Í

(33) r'H{%& ºB{(ÂØt(ÂØtBÙPLÑ&'y2x³s *' ÆÄt Á7t M G G &SLM& OYj

(34) G &('S"r')% LÑ&S"Ø G! N r') %&(' N R N RQ$$#LY%R½')L G & S*$POJ% *$Sr2x=R E! S_ÿQSr') I $DLÑ&S I LÑ$)RQ&('*%$k} I L NÑN r'RQOJ*$Stuq¤ I LM&(ÿ½RQLMRQ&(' %$]' G &S$]')LÑ' I "aS_S I %} G LM$ ! N I $ LÑ&(ÿQR½L¥R½&,'?@' $ G LÑ'LM"×"r'RQ&(') G &$)Ru*"%]m% N R½$$L=% N PÍ') E! $RQ U } G &*'L G &&S N $V I !HGJI N Qk} I L NÑN *')RQOJ{ N R I ' N R N RQ$$#LY%R½')L G &û I ') E ! $ ! LM*¼r' GJI G &S&=R½&' N ¹x G N G & Gu LÑutfÇ &Ã$ LÑ&S$ ! LÑ)RQ&('aS'S2xS&L~ I %$ N OJ" G "*'L I %$ûRQ&SR G O I %$S{ I ;t Å2¼t Á %$yxg2xS%z(s Rup G ÿ½Rm*' ÆÄ¼t Á7Tt M G G &SLÑ& bdcE OYj

(35) G &('ûR½&S& G &*" %& ¥u¦¦³¥-Rÿ G LÑ¼S#%$$#" I &S N R½$$LYR')L G &v$L L N RQLÑ !hGQI N *$&½ I f G N $ N G $n I k = 1 t GJI $ÿ GJI S#L G &S$ L~%L ! #"%$#%&(')* I &S R I ')# R !S! G 2xSEOJ" G "*'L I u¤{ S"*ÿQ N GJ!S! "% ! RQ)R NÑN N %&('r'1LM&S"/ ! %&SSR a %&('S{ ! * *'')RQ&('Y GQK ')*&SLMS*$YLÑ&(ÿQR½L¥R½&'$Q} G &r')L G &S&S N $i G ! N "*'R½&'* I ò ÁÃRQ#'LM&Sr' f R LM$tq® KSI ' N N ! LÑ&%L ! R S5**'' 'yx% $#Ã*$#'ESÔ G &S&% I &S9 RQ$$#LY%R½')L G & G ! N * ')û9*%$2x=R ! $S{ !HGJI 1' GJI ' (U9+)/B:V9 N N %$#2x=RQ&SOQ %&(')$aS GfG # G &S&S"*%$?RQ&=R s')L~ I *$St k ∈ N∗ { ¤q *$×/ Ru%LÑ&S%$_SØ**'#') "*'yx G SU$ G &('×$#LÑ' I "%%$×=RQ&S$ N %$a')RÿQR I Ã S ¤ t ¤ Sx #LM$#' GJ! Sx *S{ = °t Á RQ#SS# L~ ! $-S *' ¤ t GJI $#$%R I å =® b ¨ uvj { ¥u¦¦¦

(36) Gò N %$1R I ') I $-#"%$ G N ÿQ%&(' N R9 N R½$$LYR')L G &«S%$Ø2x=R ÿJSr') I $LM& I LM$RQ&(' I & 'k} I L NMN *'R½OJ N LÑ&S"RQLÑ={ %& ! #L RQ&(' N ! GJK N *& ')* %$× G x G f G N G OJLÑD N R½'LÑÿQut ¤ NMN f *L$#_R %&S#m N R#"%$ G N I 'L G &U I &Sa"S I R½'L G &%x G E G N G OJL~ I { *$#' f m f SLM# I &S N "S I R½'L G & S RØ} G - . ¦gt¹¦gt·¥1 X · F = G, N N 2x=R ! X RQOJLM$#$)RQ&(' ! RQ¤S"*LÑÿ½R½'L G &Y$ I %$$"%#LM*$} G NMN %$®*' G &ÿQ%#OJ%&(')*$SEt W_&U$ IS!S! N " %&('RQLÑ K G I N %$ !HGJI N R% N RQ$#$LYR½'L G & I 'R ! $S{Gm N Rur')L G & S N L RQOJ× X· } GJI &LÑ'aR N G $ I &Ã*$ ! R%YS E Ñ N N ! y*$-S%$-$zs "*'LÑ%$1 I }k I L r'RQOJ FX I L;%$ ! *'%&(' I &S } G ') E ! $US X t R {©& GJI $ N N N N I G G G K I I G G E G G I "% = LM &S$ R# R½$$LYR')L & L 'R S Z m &S;" R½')L &Øx OJL RQ$#$ G %LM"*Ûm X t *&SL% *&('H{. N M N N N s

(37) Ç#t R I ¨ {Y¥u¦¦ R-$ ISK $#'R½&'LM *&('OJ"%&"%)R LM$#"Q%*$ "r'Tx G S*$#m N RU& G R N LM$R½')L G &%} G NÑN _S%$ ! *#' I K R½'L G &$ 2xSR E ! $ ¸ÿQSr') I $< I R½$L f x G G O* &S%$ X0 { $RQ&S$Ô$ IS!!hG $#% N R«& G & f & I NÑN LÑ'" I N N N ! LÑ% v *'StDq®%$ G I *$S R[ RQ$#$LR½'L G &<} G NMN U$ LÑS%&(')L=%&('*& G #mÔ%$ ISG ')LÑ%&(')$ I & $ IS!S! N " %&(')RQLM#]m N L RQOQ×S N R1"%L ÿ½R½')L G & X0· t N N G N G OJL~ I U%$#']} Ru%L N !SI LM$y I G &ò$ R % &SRQ&=R N s f I RQ&S λ∈ / R− { "r' I SS "S I R½'L G & x G ')L~ I *&(''m N " I R½')L G & Z1 · F˜ = G˜ {¸ I L·$ "*$ GJI ' L a "%L¥R½' %&('U°L N -Ú LM$#' I &S $ G N I 'L G & $#L *' $ I N %&('a$#L G(0, t I R½&S ˜ 0) = 0 t GJI $*' GJI ÿ G &S$_R N G $ N *$"%$ I N ')R½')$Û N R½$$L I %$aS N LM&S"%RQLÑ$)R½'L G &B R X cE å c ¤ N N I G G E G G I I ! I " R½')L &Zx OJL R}"*'" "*' SLM"* RQvÁ7t %#'ySx LM* r' t *#ÿQR { u

(38) λ N G #<$y I 0 λ ÿQ"%L= +%*#')RQLM&S*$r G &SLÑ'L G &S$URQL 'Tx "*'L~ I *$1*'r I Z !HG $$n% S I & LÑ&(')"%OQ)R N ! # L~% # Gò. (0, + 0R -21& 03546/!+)78#&9;:=</ 03 9;/ 0) >#&9;3 .?@+BA C9;3 D+E')3 D:F'.

(39) jZY. SY'Õs ! N G O(RQ#LÑ'Tx L I ut! I N I *$×R½&S&S"%*$ ! N I $'RQ#{^=RQ&S$ N i%RQ$_S $# NÑN %$a#"%$ G &S&=RQ&(')*$& G & N LÑ&S"R f LM$R K N %$S{Á7t *#'yxSLM**''oÛt¸q G )RHs c C[Üj { u ¡;

(40) G &'Q G! N r') *&'LMS*&'L=" N %$ GJK $#' I r')L G &S$im "%$ GJI S# N " I R½')L G &tÇ &B=&{ N G $y I *$#'LM#)R½'L G &&S N &S"%OJR½')L} { N [%RQ$ I &e2xSR E!G K LÑ')R N *&(' N LM&"RQ#LM$)R K N URv"*')"-')RQL ')"1=R½&S$ å ®= b λ¨ uvj {D¥u¦¦¦

(41) t R RQ&S$ N R ! N IS! RQ]'PS*$'%RQ$ N *$ GJK $]') I *')L G &S$m "%$ GJI S# N " I R½')L G % & x G G N G OQ~L I G &(' !SI *')#?LÑ&(')% ! "r')"*%$D*&'% *$;S_& G & f ')#LÑÿfL¥R N LÑ'"; N RP' GJ!HG f N G OJLÑRQz$ s E! ' G ')L I P I }k I L NMN *'R½OJ?$ GJI $ f v Ru*%&('Ht ¤ *$?R !S! G 2xS*$_OQ"%&S"*)R N LM$*&('aR I G &('f')$LÑ&SO I N LM* N R "rT' x G SPS*)$ %RQ)R *')"*LÑ$#')L I %$S" %#LÑ') ! R½\ Mt=2Å t=Â×#& G N Ô=RQ&$ t[ {Y2x=R ! t=ÅÅ

(42) t W?& ! # ~L * ! RQ]')LMûÃ& G ') "*' I S $*)7 R G &S$Ru%#"%Lm¸S"r')% LÑ&S% N %$ GJK $]') I *'L G &S$S N " I R½')L G æ & x G G N G OJ~L I URQ$$ G *LM"*% m I && ½ I f G N tq G #$y I N "S I R')L G & Z · F = G RQ r' I &Sa$ G N I ')L G &rx G N GEG ! xS F { !HGJI ·' GJI '2xS LÑ& ' m Z . u %$]' f m f SLÑ G &(')*& I =RQ&$ I &S[ GJI K ûLÑ&'"%OJR N S Z 1 *' γ N 'RQ&OJ%N &(% N LMRQ&(' !HG LM&(' p0 R IÔ!HG LM&(' p1 & GJI $_Rÿ G &S$ R1 N R½'L G & Z }k I L NMN r'RQOQPRQ$$ G *LM". m. ∂ ∂ t x2 ∂x + y ∂y. F (p1 ) − F (p0 ) =. GτZ .. γ. Â×LM&$L { I &S5 G &SLÑ'L G &z&S"%*$$)R½LM¾m N ÚLM$]')%&g%ÔS F %$]' N R & I NMN LÑ')" S N LM&(')"*OJ)R N Rγ G · τZ {$ I ' GJI 'sB N γ 'RQ&SOQ%&(''m Z . $ G L ' p0 = p1 1t ¤ ! %&S=R½&' **''# G &SLÑ')L G &1*$#' LM&$ I Ô$)R½&'a$L G &-$# G &('%&(')_ LM&(')"*OJ* $ I N %$ s! N *$+ G &('%& I $û=RQ&$ N %$')R v Sr' G LÑ%$ S Z Ü%&pOJ"*&S"%R N *%$ '))R v S *' G LM#%$$ G &(' $L ! N %&('% G &&SÚ%${ = RQLM$ N L R½OJûS Z· %& ')RQ&(' I ×S"*LÑÿ½R½'L G &-*$#';SV G SL %&S$#L G &-LÑ&g=&SLÑut GJI ÜS"S*LÑ# G! N *') *&(' N L RQOJS Z· L N & GJI $¢}R I '×OJ"%&S"*)R N LM$#% N RE& G ')L G & S]2xS LM& ')RQ&SOJ*&('m W?&9s! N R½$ns E! ' G 'L~ I S x2 ∂x∂ + y ∂y∂ t % NÑN S' 2xS LM& 'RQ&OJ%&('×RQn$ s ! ' G '~L I tÄq¤R & I NMN LÑ')"P Rγ GτZ $ I a' GJI ' N N s! aRQn$ s ! ' G '~L I γ SrÿfLM%&(' I % & *LÑy' %#¼ ÚLM$]')%g& % !HGJI I &$ G I ')L G & N G u G E G ! I I G x S x û $ & ÿ M L # $ M L = & Q R Q O ûS (0, 0) t Å N *$#'E G &Ô%$#$*&'LM N SÔ$Rÿ G LMS"%LÑ N R ' GJ!hG N G OJLÑ F RQn$ s R ! ' G '~L I I }k I L NMN *'R½OJu t RQ&S$ I &SD$#S G &SS ! RQ]')LÑ;& GQI $"r' I SLÑ% G &S$ N %$ N LM*&S$©*&(')DS" I R½'L G ] & x G G N G OQL~ I *' G & v#I O(RQLM$ G & SQ2x=R ! $;S?ÿQS*' I #$StYq¤RØ N RQ$#$LYR½'L G & I ' ! $D*$#' I & ! GQK N N LÑ&S"RQLÑP! G & v#I O I * Z m U Z $Y)R %&SPLM *' %&()' m "r' I LM% N " I R½')L G + & x G G N G OJ~L I |y|. Re(x). Im(x). Z ·F =. (0,0,0). 1 −1. U. qR N RQ$$#LY%R½')L G &-S*$}k I L ÑN N r 'RQOJ*$ ! I ';R I $#$LG*'aS"%L ')a%& ')* %$ "S I R½'L G &%x GuEG N G JO L I u{ RQLÑ$ %*'#')YS%#&SL~* . ∂ X0 + R ∂y. . ·F = −. R y. %$#' ! N I $_$ ISK 'L N S ! R½ N R ! "%$#%&*US N R ! %]' I K R½')L G & R S*$×S I 9Sã½')"%$?S N "%O(R N LÑ'"ut R ! N I $S{=L N & *$#' ! RQ$a"*ÿfLM%&('× G &S&S% I &Ã$%&$S{ i} G NMN %&('H{gm I & $ G N I 'L G &ÔS**'' " I R½')L G &ò N $S G &S 1K ! I '_'y*$ K LM%&9*' "% GuEG ! xSt Ü?LM$]' G #L~ I *&'S{ N "S I R½'L G &9x GG N G OJL~ I ×%$#'×LM&(')#LM&S$nS I *&' N LÑ"%'m N R% N R½$$LYR')L G & G K L 'R N S%+ $ 2x=R E! $û ÿJ *') I $ tÄ GQI ')} G LM${D%&pOQ"%&S"*)R N { G & I ')L N LM$# N " I R½')L G &¯x GG N G OJL~ I RQ$#$ G %LÑ"%æm NMN ∂ tq® ! G %"*S"[ I û& GQI $ ! #"%$*&(' G &S$ X0 r'1& G & ! RQ$% vRQ$#$ G %LM"*ûR I 2x=R E!z! %]' I K " X0 + R ∂y N N N ~L %L;$ 1K &=R½' I *' ! I '1S$ RQ=R ! '%-*& ! RQ#'~L I LÑ% R I RQ$-%$r2x=R ! $1vÿJSr') I $ &SL N !HG ')*&'$ GJI}! N I $S"%OQ"%&S"*"%$ t ¤ N R ! * *S' {*&}$ LÑ&S$ ! LÑ)RQ&('YS*$Y'S 2xS&SL I %$ S othq G )RHs !hGQI N "S I LÑÿ½R N %&*. Ñ N N N N [ j 5

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(47) i0¤ c Ñ cBQ gj@ ikc!±k± cGª l j § jÄk± jGH Æ G LM*&(' r > 0 { θ ∈ [0, 2π[ r' 0 < δ ≤ 2π { G & "= &SL ' N ÿ>&þ #.-Õ.þ /Bgý oTpSCý A S?SLM#Sr')L G & θ f{ GJI ÿQ%#' I δ *'_S)RHs G & r V (r, θ, δ) :=. . δ (x, y) : 0 < |x| < r, |arg x − θ| < , |y| < r 2. . .. *L 'a$ GJI $ N RØ} G S R I N Ruu{ I &¾2xSR E! SPÿQS*' I #$×S'Õs ! &g½ I f G N LM& I LÑ' I &+}k I L NMN r'RQOQ N N NÑN FZ $ I rD × rD G &(' G & ! I '_S"S*LÑ RU' GJ!hG G OJLÑ I $ I - . 1 N RUS G LÑ' N N $#" ! RQR f {x = 0}NMN *$#' I &SP$" ! RQ)R')L%PN R½$$ G %LÑ"%im R ÿ½R I ! GJ! N # λ2 S Z1 tW_&S N ')L%*$#' I &Q}k I L _S I L$# ! G G &SOQ×%& I &P$ I z}Ru%RQ&SR s')L I Q GQIS! RQ&(' G #LMOJLÑ&Sut . ¥Y1 %&ûOJ"*&S"%R N L N & Ú LÑ$#') ! RQF$¼ZSP$#" ! R½)R½'L~*RQ$#$ G %LÑ"%im N RYÿ½R N I ! GJ! # tSÆÚL¸ *$#' !HGJI ]'RQ&(' N 'RQ${ G & ! I '$ IS!S!HG $* . I L ''ma2xSRQ&SOJ*×S] GfG G &S&S"%*$ N G R N %$T1¶0 I R = yR˜ r' G & SLMR I Z %$]'_SP'Õs ! J jhe@hÓ c^² cfehg t^ÆÚLÑ& G & G & SLMR I L N %$]'?'Õs ! ©iëÓHc!²^cfehg t . ³1 N %$¶k} I L NÑN *$S{gx G LM$ { $ G &('a')RQ&S$]ÿJ%#$*$Qm N RØ K R½')L G &qPÍÿQ%]')L~%R N JQ dx = 0 t {x = 0} = . µ1 !hGJI Ô $R *&' ! *'LÑ'H{ r>0$ I E . R³1 N *$ k} I L NMN *$¼S N R1#%$]')L*'L G &ÔS ! N %&(' G &S&S-Ú %$St FZ m ' GJI 'a$#Sr') I V K "$ G &('_$L E . K 1 $L %$]'_SP'Õs ! YSLÑÿQ%OQ%&('H{ N %$¶k} I L NMN %$¼ ! N *&(' G &&SÚ %$t Z FZ $ G &('?$#L . H1 $L %$#'1Sv'Õs ! + G &(ÿJ%#OJ%&('S{ N x G N G & G LM RQ$#$ G %LM"*5m N RÃS I Ú L v$" ! RQ)R')L%v" f Z ')* LM&S N R ' GJ!HG N G OJLM ¸x G LM$V%r'')-$#" ! R½)R½'L~*u{ N %$Vk} I L NMN %$_&S G &'LM*&S&S%&(' ! R½$S s! N %$P$#L µ ∈/ R tÄq G $n I N LM&(ÿ½RQLMRQ&(')} G N µ %$#'"% N { N R' GJ!HG .N G OJLM ! I 'ir ') ! N I $ GuE ! N Ú u©t IòK LM*&}R I I &Sk} I L NMN 1& %$#' $L E ! N %&(' G &S&S- µ ∈ Q r' N ¹x G N G & G f N N N LMÔ%$]' LM&S"%RQLÑ$)R K >1-{ GJImK LM*& R I ! I $ I & & Gu1 K ÔS"*& G1 K )R K N ÔS9k} I L NÑN *$-& *$#'.

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