% Jean-Pierre Demailly % Universit\'e de Grenoble I, Institut Fourier % \'Equations Diff\'erentielles et Analyse Num\'erique % Collection Grenoble-Sciences, Presses Universitaires de Grenoble \newif\ifcolorized \colorizedfalse \newif\iftocfirst \tocfirsttrue \input hyperbasics.tex \ifcolorized \magnification=1200 \hsize=13cm\hoffset=0cm \vsize=18.8cm\voffset=0.3cm \else %% \magnification=1000 \hsize=13cm\hoffset=1.5cm \vsize=18.8cm\voffset=2cm \magnification=1200 \hsize=13cm\hoffset=0cm \vsize=18.8cm\voffset=0.3cm \fi \pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \parindent=0mm \parskip=5pt plus 1pt minus 1pt \baselineskip=12.5pt \catcode`\@=11 \def\makeheadline{\vbox to\z@{\vskip-25.4\p@ \line{\vbox to8.5\p@{}\the\headline}\vss}\nointerlineskip} \def\makefootline{\baselineskip28\p@\lineskiplimit\z@\line{\the\footline}} \catcode`\@=12 % new fonts definitions \font\twentyfourss=cmss10 at 24.8832pt \font\twentyss=cmss10 at 20.736pt \font\seventeenss=cmss10 at 17.28pt \font\fourteenss=cmss10 at 14.4pt \font\twelvess=cmss10 at 12pt \font\tenss=cmss10 \font\sevenss=cmss10 at 7pt \font\twentyfourbss=cmssbx10 at 24.8832pt \font\twentybss=cmssbx10 at 20.736pt \font\seventeenbss=cmssbx10 at 17.28pt \font\fourteenbss=cmssbx10 at 14.4pt \font\twelvebss=cmssbx10 at 12pt \font\tenbss=cmssbx10 \font\sevenbss=cmssbx10 at 7pt \font\twelveline=line10 at 12pt \font\twentyrm=cmr10 at 20.736pt \font\seventeensy=cmsy10 at 17.28pt \font\seventeenrm=cmr10 at 17.28pt \font\seventeenbf=cmbx10 at 17.28pt \font\seventeeni=cmmi10 at 17.28pt \font\fourteensy=cmsy10 at 14.4pt \font\fourteenrm=cmr10 at 14.4pt \font\fourteenbf=cmbx10 at 14.4pt \font\fourteeni=cmmi10 at 14.4pt \font\twelvesy=cmsy10 at 12pt \font\twelvebf=cmbx10 at 12pt \font\twelvei=cmmi10 at 12pt \font\eightbf=cmbx10 at 8pt \font\sixbf=cmbx10 at 6pt \font\twelvei=cmmi10 at 12pt \font\eighti=cmmi10 at 8pt \font\sixi=cmmi10 at 6pt \font\twelverm=cmr10 at 12pt \font\eightrm=cmr10 at 8pt \font\sixrm=cmr10 at 6pt \font\eightsy=cmsy8 \font\sixsy=cmsy6 \font\eightit=cmti8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\seventeenbsy=cmbsy10 at 17.28pt \font\fourteenbsy=cmbsy10 at 14.4pt \font\twelvebsy=cmbsy10 at 12pt \font\tenbsy=cmbsy10 \font\eightbsy=cmbsy8 \font\sevenbsy=cmbsy7 \font\sixbsy=cmbsy6 \font\fivebsy=cmbsy5 \font\tenmsa=msam10 \font\eightmsa=msam8 \font\sevenmsa=msam7 \font\fivemsa=msam5 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \def\msa{\fam\msafam\tenmsa} \font\fourteenmsb=msbm10 at 14.4pt \font\tenmsb=msbm10 \font\eightmsb=msbm8 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb{\fam\msbfam\tenmsb} \font\fourteenCal=eusm10 at 14.4pt \font\twelveCal=eusm10 at 12pt \font\tenCal=eusm10 \font\sevenCal=eusm7 \font\fiveCal=eusm5 \newfam\Calfam \textfont\Calfam=\tenCal \scriptfont\Calfam=\sevenCal \scriptscriptfont\Calfam=\fiveCal \def\Cal{\fam\Calfam\tenCal} \font\teneuf=eusm10 \font\teneuf=eufm10 \font\seveneuf=eufm7 \font\fiveeuf=eufm5 \newfam\euffam \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \def\euf{\fam\euffam\teneuf} \font\seventeenbfit=cmmib10 at 17.28pt \font\fourteenbfit=cmmib10 at 14.4pt \font\twelvebfit=cmmib10 at 12pt \font\tenbfit=cmmib10 \font\eightbfit=cmmib8 \font\sevenbfit=cmmib7 \font\sixbfit=cmmib6 \font\fivebfit=cmmib5 \newfam\bfitfam \textfont\bfitfam=\tenbfit \scriptfont\bfitfam=\sevenbfit \scriptscriptfont\bfitfam=\fivebfit \def\bfit{\fam\bfitfam\tenbfit} \def\gS{\hbox{\teneuf S}} % changing font sizes \catcode`\@=11 \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\mit{\fam\@ne\eighti}% \def\oldstyle{\fam\@ne\eighti}% \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\msbfam=\eightmsb \def\Bbb{\fam\msbfam\eightmsb}% \abovedisplayskip=9pt plus 2pt minus 6pt \abovedisplayshortskip=0pt plus 2pt \belowdisplayskip=9pt plus 2pt minus 6pt \belowdisplayshortskip=5pt plus 2pt minus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm \normalbaselines\rm} \catcode`\@=12 \def\eightpointbf{% \textfont0=\eightbf \scriptfont0=\sixbf \scriptscriptfont0=\fivebf \textfont1=\eightbfit \scriptfont1=\sixbfit \scriptscriptfont1=\fivebfit \textfont2=\eightbsy \scriptfont2=\sixbsy \scriptscriptfont2=\fivebsy \eightbf \baselineskip=10pt} \def\tenpointbf{% \textfont0=\tenbf \scriptfont0=\sevenbf \scriptscriptfont0=\fivebf \textfont1=\tenbfit \scriptfont1=\sevenbfit \scriptscriptfont1=\fivebfit \textfont2=\tenbsy \scriptfont2=\sevenbsy \scriptscriptfont2=\fivebsy \bf} \def\twelvepointbf{% \textfont0=\twelvebf \scriptfont0=\eightbf \scriptscriptfont0=\sixbf \textfont1=\twelvebfit \scriptfont1=\eightbfit \scriptscriptfont1=\sixbfit \textfont2=\twelvebsy \scriptfont2=\eightbsy \scriptscriptfont2=\sixbsy \twelvebss \baselineskip=14.4pt} \def\fourteenpointbf{% \textfont0=\fourteenbf \scriptfont0=\tenbf \scriptscriptfont0=\sevenbf \textfont1=\fourteenbfit \scriptfont1=\tenbfit \scriptscriptfont1=\sevenbfit \textfont2=\fourteenbsy \scriptfont2=\tenbsy \scriptscriptfont2=\sevenbsy \fourteenbss \baselineskip=17.28pt} \def\seventeenpointbf{% \textfont0=\seventeenbf \scriptfont0=\twelvebf \scriptscriptfont0=\eightbf \textfont1=\seventeenbfit\scriptfont1=\twelvebfit\scriptscriptfont1=\eightbfit \textfont2=\seventeenbsy \scriptfont2=\twelvebsy \scriptscriptfont2=\eightbsy \seventeenbss \baselineskip=20.736pt} \def\twelvepoint{% \textfont0=\twelverm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm \textfont1=\twelvei \scriptfont1=\eighti \scriptscriptfont1=\sixi \textfont2=\twelvesy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy \twelvess \baselineskip=14.4pt} \def\fourteenpoint{% \textfont0=\fourteenrm \scriptfont0=\tenrm \scriptscriptfont0=\sevenrm \textfont1=\fourteeni \scriptfont1=\teni \scriptscriptfont1=\seveni \textfont2=\fourteensy \scriptfont2=\tensy \scriptscriptfont2=\sevensy \fourteenss \baselineskip=17.28pt} \def\seventeenpoint{% \textfont0=\seventeenrm \scriptfont0=\twelverm \scriptscriptfont0=\eightrm \textfont1=\seventeeni \scriptfont1=\twelvei \scriptscriptfont1=\eighti \textfont2=\seventeensy \scriptfont2=\twelvesy \scriptscriptfont2=\eightsy \seventeenss \baselineskip=20.736pt} \def\twentypoint{% \textfont0=\twentyrm \scriptfont0=\fourteenrm \scriptscriptfont0=\eightrm \textfont1=\twentyi \scriptfont1=\fourteeni \scriptscriptfont1=\eighti \textfont2=\twentysy \scriptfont2=\fourteensy \scriptscriptfont2=\eightsy \twentyss \baselineskip=24.8832pt} % main item macros \newdimen\srdim \srdim=\hsize \newdimen\irdim \irdim=\hsize \def\NOSECTREF#1{\noindent\hbox to \srdim{\null\dotfill ???(#1)}} \def\SECTREF#1{\noindent\hbox to \srdim{\csname REF\romannumeral#1\endcsname}} \def\INDREF#1{\noindent\hbox to \irdim{\csname IND\romannumeral#1\endcsname}} \newlinechar=`\^^J \def\openauxfile{ \immediate\openin1\jobname.aux \ifeof1 \message{^^JCAUTION\string: you MUST run TeX a second time^^J} \let\sectref=\NOSECTREF \let\indref=\NOSECTREF \else \input \jobname.aux \message{^^JCAUTION\string: if the file has just been modified you may have to run TeX twice^^J} \let\sectref=\SECTREF \let\indref=\INDREF \fi \message{to get correct page numbers displayed in Contents or Index Tables^^J} \immediate\openout1=\jobname.aux \let\END=\end \def\end{\immediate\closeout1\END}} \newcount\notthispage \notthispage=1 \newbox\titlebox \setbox\titlebox\hbox{\hfil} \newbox\sectionbox \setbox\sectionbox\hbox{\hfil} \newbox\chapterbox \setbox\chapterbox\hbox{\hfil} \newbox\bookbox \setbox\bookbox\hbox{\hfil} \newbox\ebookbox \setbox\ebookbox\hbox{\hfil} \def\fakeskip{$\rlap{\phantom{\vrule height18pt depth2pt width0pt}}$} \def\blackline{\rlap{\vrule width \hsize height -4.7pt depth 5pt}} \def\upperblackline{\rlap{\vrule width \hsize height 11pt depth -10.7pt}} \def\foliotoc{\ifnum\pageno=\notthispage \hfil \else \ifodd\pageno \blackline% {\tenss \copy\chapterbox% \hfill\romannumeral\pageno}\else\blackline% {\tenss\romannumeral\pageno\hfill\copy\titlebox}\fi\fi} \def\foliochap{\ifnum\pageno=\notthispage \hfil \else \ifodd\pageno \blackline% {\tenss \copy\sectionbox% \hfill\number\pageno}\else\blackline% {\tenss\number\pageno\hfill\copy\chapterbox}\fi\fi} \def\folioref{\ifnum\pageno=\notthispage \hfil \else \ifodd\pageno \blackline% {\tenss \copy\chapterbox% \hfill\number\pageno}\else\blackline% {\tenss\number\pageno\hfill\copy\titlebox}\fi\fi} \def\foliopapebook{\blackline{\copy\bookbox\hfill\copy\ebookbox}} \footline={\hfil} \let\forceheader\eject \def\blankline{\phantom{}\hfil\vskip0pt} \def\chapterspacing{\phantom{$\ $}\vskip2.5cm} \def\titlerunning#1{\setbox\titlebox\hbox{\tenss #1}} \def\chapterrunning#1{\notthispage=\pageno \setbox\chapterbox\hbox{\tenss #1}} \def\title#1{\noindent\hfil$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\titleleft#1{\noindent$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\supersection#1{% \par\vskip1.5cm\penalty -100 \vbox{\baselineskip=17.28pt\noindent{{\fourteenpointbf #1}}} \vskip3pt \penalty 500 \titlerunning{#1}} \newcount\numbersection \numbersection=-1 \def\sectionrunning#1{\setbox\sectionbox\hbox{\tenss #1} \write1{\string\def\string\REF \romannumeral\numbersection\string{% \noexpand#1 \string\dotfill\string\space\number\pageno\string}}} \def\RGBColor#1#2{\special{color push rgb #1}#2\special{color pop}} \ifcolorized \def\maincolor#1{\RGBColor{0 0 0.5}{#1}} \else \def\maincolor#1{#1} \fi \def\colorstate#1{\noindent\maincolor{{\bf #1.}}} \def\section#1{% \removelastskip \vskip1cm\penalty -100 \vbox{\baselineskip=17.28pt\noindent{\maincolor{\seventeenpoint #1\vskip0pt}}} \vskip1pt \penalty 500 \advance\numbersection by 1 \sectionrunning{#1}} \def\subsection#1{% \removelastskip \vskip0.5cm\penalty -100 \vbox{\noindent{\maincolor{\fourteenpoint #1\vskip0pt}}} \penalty 500} \newcount\numberindex \numberindex=0 \def\index#1#2{% \advance\numberindex by 1 \write1{\string\def \string\IND #1% \romannumeral\numberindex \string{% \noexpand#2 \string\dotfill \space \number\numbersection, p.\string\ \space\number\pageno \string}}} \newcount\numberchap \iftocfirst \numberchap=-2 \else \numberchap=-1 \fi \def\CHpage{ \immediate\write1{\string\def\string\CH \romannumeral\numberchap\string{% \number\pageno\string}}} \def\chapterjump{ \vfill\eject \ifodd\pageno \else {\headline={\hfil}\null\vskip0pt\vfill\eject} \fi \advance\numberchap by 1 \CHpage } \def\numpage#1{~\dotfill~#1} \newdimen\dpp \newbox\claimbox \setbox\claimbox\hbox{\hfil} \long\def\claim#1#2\endclaim{\par\vskip 5pt\noindent {{\tenpointbf\maincolor{#1.}}\ {\it #2}\vskip-18pt} \strut\kern\hsize \ifcolorized \special{" gsave 0.8 0.8 0.1 setrgbcolor -1 -1 scale 0 6 moveto 7 0 rlineto 0 -3 rlineto -4 0 rlineto 0 -4 rlineto -3 0 rlineto closepath fill grestore}% \else \special{" gsave 0.667 setgray -1 -1 scale 0 6 moveto 7 0 rlineto 0 -3 rlineto -4 0 rlineto 0 -4 rlineto -3 0 rlineto closepath fill grestore}% \fi \par\vskip11pt} \long\def\exo#1#2\endexo{\par\vskip 5pt\noindent {\tenpointbf\maincolor{#1.}}\ {\rm #2}\par} \def\ceqno(#1){\eqno{{\maincolor{(#1)}}}} \def\cleqno(#1){\leqno{{\maincolor{(#1)}}}} \def\eqleqno#1\hfill#2{\leqno \hbox to 0.0001pt{\rlap{\rlap{\maincolor{#1}}% \kern\hsize\llap{\maincolor{#2}}}}} \def\joinrel{\mathrel{\mkern-3.5mu}} \def\llra{\relbar\joinrel\relbar\joinrel\longrightarrow} \def\llmapsto{\mapstochar\relbar\joinrel\relbar\joinrel\longrightarrow} \def\vlra#1|{\mathrel{\hbox to#1mm{\rightarrowfill}}} % Usual sets of numbers \def\bC{{\Bbb C}} \def\bK{{\Bbb K}} \def\bN{{\Bbb N}} \def\bP{{\Bbb P}} \def\bQ{{\Bbb Q}} \def\bR{{\Bbb R}} \def\bZ{{\Bbb Z}} % Calligraphic capital letters \def\cA{{\Cal A}} \def\cC{{\Cal C}} \def\cD{{\Cal D}} \def\cE{{\Cal E}} \def\cF{{\Cal F}} \def\cG{{\Cal G}} \def\cI{{\Cal I}} \def\cL{{\Cal L}} \def\cN{{\Cal N}} \def\cP{{\Cal P}} \def\cR{{\Cal R}} \def\cS{{\Cal S}} \def\cT{{\Cal T}} \def\cV{{\Cal V}} \def\tC{\smash{\tilde C}} \def\tvi{{\vrule height 10pt depth 5pt width 0pt}} \def\tv{\tvi\vrule} \def\hexnbr#1{\ifnum#1<10 \number#1\else \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi} \def\msatype{\hexnbr\msafam} \def\msbtype{\hexnbr\msbfam} \mathchardef\smallsetminus="2\msbtype72 \let\ssm\smallsetminus \mathchardef\restr="3\msatype16 \mathchardef\supsetneqq="3\msbtype25 \mathchardef\subsetneq="3\msbtype28 \mathchardef\supsetneq="3\msbtype29 \mathchardef\leqslant="3\msatype36 \mathchardef\geqslant="3\msatype3E \mathchardef\complement="0\msatype7B \let\ge=\geqslant \let\le=\leqslant \let\dsp=\displaystyle \def\VVert{{|}\kern-1.2pt{|}\kern-1.2pt{|}} \def\square{\maincolor{\hfill \hbox{ \vrule height 1.453ex width 0.093ex depth 0ex \vrule height 1.5ex width 1.3ex depth -1.407ex\kern-0.1ex \vrule height 1.453ex width 0.093ex depth 0ex\kern-1.35ex \vrule height 0.093ex width 1.3ex depth 0ex}}} \def\qed{\phantom{$\quad$}$\square$} \def\frac#1#2{{#1\over #2}} \def\dfrac#1#2{{#1\over #2}} \def\fracs#1#2{{\scriptstyle{#1\over#2}}} \def\fracss#1#2{{\scriptscriptstyle{#1\over#2}}} \catcode`@=11 \def\ccmalign#1#2{\null\,\vcenter{\normalbaselines\m@th \ialign{$\displaystyle ##$\hfil&&\kern#1$\displaystyle ##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #2\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \def\cmalign#1{\null\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$&&$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \catcode`@=12 \def\coloritem#1{\item{\maincolor{#1}}} \def\itemv{% \item{\llap{$\square\kern7pt$}\llap{$\raise0.6pt\hbox{$\times$}\kern6pt$}}} \def\itemf{% \item{\llap{$\square\kern7pt$}}} \def\bul{$\scriptstyle\bullet$} \def\bigzero{\hbox{~\seventeenrm 0~~~}} \def\branch{\raise2pt\hbox{ \vbox{\hbox{\vrule height 0.4pt depth 0pt width 12.2pt} \hbox{\raise-11.6pt\hbox{\twelveline\char"00}}}}} \def\?{\hbox{$\,$}} \def\lguil{\hbox{% \raise1pt\hbox{$\scriptscriptstyle\langle\!\langle\kern1pt$}}} \def\rguil{\hbox{% \raise1pt\hbox{$\kern1pt\scriptscriptstyle\rangle\!\rangle$}}} \def\sqind#1{\kern1.5pt\rlap{\raise5pt\hbox{$\scriptstyle#1$}}\kern-1.5pt} \def\KH{{\rm KH}} \def\Riemann{{\rm Riemann}} \def\eq{\mathop{\rm =}} \def\Bij{\mathop{\rm Bij}} \def\tr{\mathop{\rm tr}} \def\Log{\mathop{\rm Log}} \def\Mat{\mathop{\rm Mat}} \def\Ker{\mathop{\rm Ker}} \def\Vect{\mathop{\rm Vect}} \def\rang{\mathop{\rm rang}\nolimits} \def\Inv{\mathop{\rm Inv}} \def\Sp{\mathop{\rm Sp}\nolimits} \def\Arccos{\mathop{\rm Arccos}} \def\Arcsin{\mathop{\rm Arcsin}} \def\Arctan{\mathop{\rm Arctan}} \def\Argcosh{\mathop{\rm Argcosh}} \def\Argsinh{\mathop{\rm Argsinh}} \def\cotan{\mathop{\rm cotan}} \def\cotanh{\mathop{\rm cotanh}} \def\supess{\mathop{\rm sup\,ess}} \def\infess{\mathop{\rm inf\,ess}} \def\card{\mathop{\rm card}} \def\Id{\mathop{\rm Id}\nolimits} \def\Re{\mathop{\rm Re}} \def\Im{\mathop{\rm Im}} \def\Jac{\mathop{\rm Jac}} \def\Supp{\mathop{\rm Supp}} \def\oscil{\mathop{\rm oscil}} \def\vol{\mathop{\rm vol}} \def\aspect{\mathop{\rm aspect}} \def\aire{\mathop{\rm aire}} \def\longueur{\mathop{\rm longueur}} \def\diam{\mathop{\rm diam}} \def\sign{\mathop{\rm sign}} \def\grad{\mathop{\rm grad}} \def\rot{\mathop{\rm rot}} \def\div{\mathop{\rm div}} \def\Var{\mathop{\rm Var}} \def\pf{p\kern-1pt f} \def\dplus{\mathrel{\dot +}} \def\note#1#2#3{\footnote{}% {\baselineskip=8pt\leftskip=5.4mm\rlap{\strut\kern-5.4mm${}^{\rm(#2)}$}% \hyperdef#1{}{}{}{\eightpoint #3}\vskip-15pt}% ${}^{\rm(\hyperref#1{#2})}$} \def\\{\hfil\break} \def\demi{\textstyle{1\over 2}} \def\ovl{\overline} \def\ovr{\overrightarrow} \def\ul#1{$\underline{\smash{\hbox{#1}}}$} \def\build#1^#2_#3{\mathop{#1}\limits^{#2}_{#3}} \def\demo{\noindent{\it D\'emonstration.}\ } \newbox\formulabox \setbox\formulabox\hbox{\hfil} \newdimen\wdd \newdimen\htt \def\boxed#1#2{\setbox\formulabox\hbox{$\displaystyle #2$} \wdd=8pt \advance \wdd by \wd\formulabox \htt=4pt \advance \htt by \ht\formulabox \dpp=4pt \advance \dpp by \dp\formulabox \ifcolorized \smash{\rlap{\hbox{\RGBColor{1 0.92 0.94}{\kern-4pt\vrule width \wdd height \htt depth \dpp}}}}\else% \smash{\rlap{\hbox{\RGBColor{0.94 0.94 0.94}{\kern-4pt\vrule width \wdd height \htt depth \dpp}}}}\fi% \copy\formulabox} \def\boxit#1#2{\hbox{\vrule \vbox{\hrule\kern#1 \vtop{\hbox{\kern#1 #2\kern#1}% \kern#1\hrule}}% \vrule}} \def\boxmat#1#2{\boxit{#1}{$#2$}} % inclusion of PostScript files \special{header=mdrlib.ps} \long\def\InsertFig#1 #2 #3 #4\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" #3}}#4$}} \long\def\InsertPSFile#1 #2 #3 #4 #5 #6\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile=#5 hscale=#3 vscale=#4}}#6$}} \long\def\InsertImage#1 #2 #3 #4 #5 #6 #7 #8\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile="`img2eps file #7 height #4 mm width #3 mm gamma #5 angle #6}}#8$}} \long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}} \def\psrotate#1#2{\special{ps: /a{pop pop}def gsave #1 neg rotate}#2 \special{ps: grestore /a{moveto}bind def}} % bibibliography \def\bibitem#1&% {\hangindent=1.2cm\hangafter=1 \noindent\rlap{\hbox{\eightpointbf #1}}\kern1.2cm{\rm #2}{\it #3}{\rm #4.}} %%% Texte principal \setbox\bookbox\hbox{\tenss \maincolor{% Analyse num\'erique et \'equations diff\'erentielles}} \setbox\ebookbox\hbox{\tenss \maincolor{% \href{https://grenoble-sciences.ujf-grenoble.fr/pap-ebook/demailly/1-calculs-numeriques-approches}{\RGBColor{0 0 1}{$\strut\,\Rightarrow$ site web compagnon$\,$}}}} \footline={\upperblackline \maincolor{% \tenss Le proc\'ed\'e de sommation d'Euler\hfill\copyright\ Grenoble Sciences, J.-P.~Demailly $\scriptstyle \bullet$ \folio}} \openauxfile \headline={\foliopapebook} \notthispage=\pageno \strut\vskip4mm \maincolor{% \title{Le proc\'ed\'e de sommation d'Euler} \titlerunning{Le proc\'ed\'e de sommation d'Euler}} \bigskip\bigskip Soit $a>0$ un r\'eel, et $(u_n)_{n\ge 0}$ une suite r\'eelle ou complexe. On lui associe la suite $(\tilde u_n)$ telle que $$ \tilde u_n={1\over (1+a)^n}\sum_{k=0}^n{n\choose k}a^ku_k \leqno\hyperdef\EQSEi{}{}{({\rm SE}1)} $$ (le proc\'ed\'e de sommation d'Euler le plus classique correspond au choix $a=1$). {\bf Th\'eor\`eme.} {\it Si la suite $(u_n)$ converge vers une limite $\ell$, alors la suite $(\tilde u_n)$ converge \'egalement vers $\ell$.} {\it D\'emonstration.} Comme $\sum_{k=0}^n{n\choose k}a^k=(1+a)^n$, on a $$ \tilde u_n-\ell={1\over (1+a)^n}\sum_{k=0}^n{n\choose k}a^k(u_k-\ell) \quad\Longrightarrow\quad \big|\tilde u_n-\ell\big|\le {1\over (1+a)^n}\sum_{k=0}^n{n\choose k}a^k\big|u_k-\ell\big|. $$ Soit $M$ un majorant de $|u_n|$, de sorte que $|\ell|\le M$ et $|u_n-\ell|\le 2M$. \'Etant donn\'e $\varepsilon>0$, il existe un rang $N$ tel que pour tout $k\ge N$ on ait $|u_k-\ell|\le\varepsilon$. D\'ecoupons la sommation en les indices $k\le N-1$ et $k\ge N$, en supposant $n\ge N$. Nous obtenons $$ \eqalign{ {1\over (1+a)^n}\sum_{k=0}^{n}{n\choose k}a^k\big|u_k-\ell\big| &\le{1\over (1+a)^n}\bigg(\sum_{k=0}^{N-1}{n\choose k}a^k\times 2M +\sum_{k=N}^n{n\choose k}a^k\varepsilon\bigg)\cr &\le\varepsilon+2M{1\over (1+a)^n}\sum_{k=0}^{N-1}{n\choose k}a^k.\cr } $$ Pour $0<\delta<1$ (par exemple $\delta=1/2$), on voit par ailleurs que $$ {1\over (1+a)^n}\sum_{k=0}^{N-1}{n\choose k}a^k \le{\delta^{1-N}\over (1+a)^n}\sum_{k=0}^{N-1}{n\choose k}(\delta a)^k \le \delta^{1-N}\Big({1+\delta a\over 1+a}\Big)^n $$ tend vers $0$ quand $n\to +\infty$, donc on peut choisir $N_1>N$ tel que $n\ge N_1$ implique $$ {1\over (1+a)^n}\sum_{k=0}^{N-1}{n\choose k}a^k\le {\varepsilon\over 2M}. $$ Ceci montre que $|\tilde u_n-\ell|\le 2\varepsilon$ pour $n\ge N_1$, et le th\'eor\`eme est d\'emontr\'e.\qed \medskip Pour transformer une s\'erie $\sum_{n\ge 0}u_n$, on applique le proc\'ed\'e pr\'ec\'edent \`a la suite des sommes partielles $$ s_n=\sum_{0\le k0$. \medskip {\bf Exemple 1.} On consid\`ere la s\'erie enti\`ere du logarithme n\'ep\'erien $$ \ln(1+x)=\sum_{n=0}^{+\infty}(-1)^n{x^{n+1}\over n+1},\qquad x\in{}]0,1]. $$ En choisissant $a={1\over x}$, on obtient $$ \eqalign{ \tilde u_n&={1/x\over (1+1/x)^{n+1}}\sum_{k=0}^n{n\choose k}{(-1)^k\over k+1}x ={1\over (1+1/x)^{n+1}}\int_0^1\sum_{k=0}^n{n\choose k}(-1)^kt^k\,dt\cr &={1\over (1+1/x)^{n+1}}\int_0^1(1-t)^n\,dt ={1\over (n+1)(1+1/x)^{n+1}}.\cr} $$ On trouve l'identit\'e $$ \sum_{n=0}^{+\infty}(-1)^n{x^{n+1}\over n+1}= \sum_{n=0}^{+\infty}{1\over (n+1)(1+1/x)^{n+1}}, $$ qui \'equivaut \`a $\ln(1+x)=-\ln(1-{1\over 1+1/x})$. On observera que le membre de droite am\'eliore tr\`es notablement la convergence de la s\'erie, en particulier pour $x=1$ (on passe d'une convergence lente \`a une convergence g\'eom\'etrique). \medskip {\bf Exemple 2.} On consid\`ere la s\'erie enti\`ere de la fonction $\Arctan$ $$ \Arctan x=\sum_{n=0}^{+\infty}(-1)^n{x^{2n+1}\over 2n+1},\qquad x\in{}]0,1]. $$ En choisissant $a={1\over x^2}$, on obtient $$ \eqalign{ \tilde u_n &={1/x^2\over (1+1/x^2)^{n+1}}\sum_{k=0}^n{n\choose k}{(-1)^k\over 2k+1}x\cr &={1/x\over (1+1/x^2)^{n+1}}\int_0^1\sum_{k=0}^n{n\choose k}(-1)^kt^{2k}\,dt ={1/x\over (1+1/x^2)^{n+1}}\int_0^1(1-t^2)^n\,dt.\cr} $$ La valeur de l'int\'egrale $I_n=\int_0^1(1-t^2)^n\,dt= \int_0^{\pi/2}\cos^{2n+1}t\,dt$ est un grand classique (cf.\ formule de Wallis), on trouve la relation de r\'ecurrence $I_n={2n\over 2n+1}I_{n-1}$ et il en r\'esulte $$ I_0=1,\qquad I_n={2\cdot 4\cdot\cdots\cdot 2n\over 1\cdot 3\cdot 5\cdot\cdots\cdot 2n+1}\quad \hbox{pour $n\ge 1$}. $$ On obtient l'identit\'e int\'eressante $$ \Arctan x= {1\over x}~\sum_{n=0}^{+\infty}{1\over (1+1/x^2)^{n+1}} {2\cdot 4\cdot\cdots\cdot 2n\over 1\cdot 3\cdot 5\cdot\cdots\cdot 2n+1}, \leqno\hyperdef\EQSEiii{}{}{({\rm SE}3)} $$ et en particulier, il r\'esulte des \'egalit\'es $\Arctan(1)={\pi\over 4}$ et $\Arctan{1\over \sqrt{3}}={\pi\over 6}$ les formules tr\`es simples suivantes d\'ecouvertes par Euler~: $$\eqalign{ \pi&=2~\sum_{n=0}^{+\infty}{n!\over 1\cdot 3\cdot 5\cdot\cdots\cdot 2n+1},\cr \pi&={3\sqrt{3}\over 2}~\sum_{n=0}^{+\infty}{2^{-n}\,n!\over 1\cdot 3\cdot 5\cdot\cdots\cdot 2n+1},\cr} $$ dont la convergence est approximativement en $2^{-n}$, resp.\ $4^{-n}$. On peut bien s\^ur aussi combiner (\hyperref\EQSEiii{SE3}) avec les formules du type de celle de John Machin $$ \pi=16\,\Arctan{1\over 5}-4\,\Arctan{1\over 239}. $$ En utilisant un argument de prolongement analytique pour les fonctions $\bR$-analy\-tiques, il n'est pas difficile de voir que (\hyperref\EQSEiii{SE3}) reste vrai pour tout $x>0$. \vfill\eject {\bf Code pour PARI/GP~:} Initialiser par {\tt u=3*sqrt(3)/2;s=u;n=0; } puis it\'erer la ligne {\tt n=n+1;u=n*u/(4*n+2);s=s+u} \end % Local Variables: % TeX-command-default: "uTeX" % End: