MesCommandes.sty

\NeedsTeXFormat{LaTeX2e}
 
%\ProvidesPackage{MesCommandes}[03/03/2009 commandes usuelles]
 
% extensions
 
%\RequirePackage[latin1]{inputenc}
%\RequirePackage[T1]{fontenc}
%\RequirePackage{lmodern}
%\RequirePackage{graphicx}
%\RequirePackage[french]{babel}
%\RequirePackage{color}
%\RequirePackage{footnpag}
%\RequirePackage{amsthm}
%\RequirePackage{amsmath}
%\RequirePackage{amsfonts}
%\RequirePackage{amssymb}
%\RequirePackage{fullpage}
%\RequirePackage{graphicx}
%\RequirePackage{graphics}


% pour faire des fleches type left-right-arrow avec du texte par dessus.
% codes repris du Package extarrows-test
\def\xleftrightarrowfill@{\arrowfill@\leftarrow\relbar\rightarrow}
\providecommand{\xleftrightarrow}[2][]{%
	\ext@arrow 0099\xleftrightarrowfill@{#1}{#2}}

% commandes personnelles
 
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\makeatletter
%\renewcommand\theequation{\thesection.\arabic{equation}}
%\@addtoreset{equation}{section}
%\makeatother
%

%%CALDEROn
\newcommand{\Utt}{\mathtt{U}}
\newcommand{\Uttps}{\Utt^{+}}
\newcommand{\Uttpps}{\Utt^{+}_p}
\newcommand{\Uttm}{\Utt^{-}}
\newcommand{\Uttmp}{\Utt^{-}_p}
\newcommand{\Uttinc}{\Utt^{\rm inc}}
\newcommand{\Uttincp}{\Utt^{\rm inc}_p}

   
 %NumŽrotation liŽe
\newtheorem{thm}{Theorem}[chapter]
\newtheorem{theoreme}[thm]{ThŽorme}
\newtheorem{theorem}[thm]{Theorem}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{lemme}[thm]{Lemme}
\newtheorem{pro}[thm]{Property}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{rmq}[thm]{Remark}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{remark}[thm]{Remark}
\newtheorem{corollaire}[thm]{Corollaire}
\newcommand{\rfb}[1]{\mbox{\rm
   (\ref{#1})}\ifx\undefined\stillediting\else:\fbox{$#1$}\fi}
%
%\newtheorem{theorem}[thm]{Theorem}
%\newtheorem{proposition}[thm]{Property}
%\newtheorem{corollary}[thm]{Corollary}
%\newtheorem{remark}[thm]{Remark}
%

\newcommand{\ie}{\textit{i.e.}\xspace}
\newcommand{\eg}{\textit{e.g.}\xspace}
\newcommand{\ssi}{if and only if\xspace}

%% MU DIFF
\newcommand{\mudiff}{$\mu$-diff\xspace}
\newcommand{\matlab}{Matlab\xspace}
\newcommand{\Nscat}{M}

\newcommand{\kint}{k^-}
\newcommand{\kintp}{\kint_p}
\newcommand{\kintq}{\kint_q}


\newcommand{\FDN}{F_{DN}(\Omegam)}
\newcommand{\FN}{F_{N}(\Omegam)}
\newcommand{\FD}{F_{D}(\Omegam)}

\newcommand{\um}{u^-}
\newcommand{\ups}{u^+}

\newcommand{\ump}{u^-_p}


\newcommand{\rhomi}{\rho^-}
\newcommand{\rhops}{\rho^+}

%TEXTCOLOR

\newcommand{\red}[1]{\textcolor{red}{#1}}

% \MATHBB
\newcommand{\Ab} {{\mathbb{A}}}
\newcommand{\Bb} {{\mathbb{B}}}
\newcommand{\Cb} {{\mathbb{C}}}
\newcommand{\Db} {{\mathbb{D}}}
\newcommand{\Eb} {{\mathbb{E}}}
\newcommand{\Fb} {{\mathbb{F}}}
\newcommand{\Hb} {{\mathbb{H}}}
\newcommand{\Ib} {{\mathbb{I}}}
\newcommand{\Jb} {{\mathbb{J}}}
\newcommand{\Kb} {{\mathbb{K}}}
\newcommand{\Lb} {{\mathbb{L}}}
\newcommand{\Mb} {{\mathbb{M}}}
\newcommand{\Nb} {{\mathbb{N}}}
\newcommand{\Pb} {{\mathbb{P}}}
\newcommand{\Qb} {{\mathbb{Q}}}
\newcommand{\Rb} {{\mathbb{R}}}
\newcommand{\Sb} {{\mathbb{S}}}
\newcommand{\Tb} {{\mathbb{T}}}
\newcommand{\Ub} {{\mathbb{U}}}
\newcommand{\Yb} {{\mathbb{Y}}}
\newcommand{\Zb} {{\mathbb{Z}}}

% \scr
\newcommand{\Ascr} {{\mathscr A}}
\newcommand{\Bscr} {{\mathscr B}}
\newcommand{\Cscr} {{\mathscr C}}
\newcommand{\Dscr} {{\mathscr D}}
\newcommand{\Escr} {{\mathscr E}}
\newcommand{\Fscr} {{\mathscr F}}
\newcommand{\Gscr} {{\mathscr G}}
\newcommand{\Hscr} {{\mathscr H}}
\newcommand{\Iscr} {{\mathscr I}}
\newcommand{\Jscr} {{\mathscr J}}
\newcommand{\Kscr} {{\mathscr K}}
\newcommand{\Lscr} {{\mathscr L}}
\newcommand{\Mscr} {{\mathscr M}}
\newcommand{\Nscr} {{\mathscr N}}
\newcommand{\Oscr} {{\mathscr O}}
\newcommand{\Pscr} {{\mathscr P}}
\newcommand{\Qscr} {{\mathscr Q}}
\newcommand{\Rscr} {{\mathscr R}}
\newcommand{\Sscr} {{\mathscr S}}
\newcommand{\Tscr} {{\mathscr T}}
\newcommand{\Uscr} {{\mathscr U}}
\newcommand{\Vscr} {{\mathscr V}}
\newcommand{\Wscr} {{\mathscr W}}

% \CAL
\newcommand{\Acal} {{\cal A}}
\newcommand{\Bcal} {{\cal B}}
\newcommand{\Ccal} {{\cal C}}
\newcommand{\Dcal} {{\cal D}}
\newcommand{\Ecal} {{\cal E}}
\newcommand{\Fcal} {{\cal F}}
\newcommand{\Gcal} {{\cal G}}
\newcommand{\Hcal} {{\cal H}}
\newcommand{\Ical} {{\cal I}}
\newcommand{\Jcal} {{\cal J}}
\newcommand{\Kcal} {{\cal K}}
\newcommand{\Lcal} {{\cal L}}
\newcommand{\Mcal} {{\cal M}}
\newcommand{\Ncal} {{\cal N}}
\newcommand{\Ocal} {{\cal O}}
\newcommand{\Pcal} {{\cal P}}
\newcommand{\Qcal} {{\cal Q}}
\newcommand{\Rcal} {{\cal R}}
\newcommand{\Scal} {{\cal S}}
\newcommand{\Tcal} {{\cal T}}
\newcommand{\Ucal} {{\cal U}}
\newcommand{\Vcal} {{\cal V}}
\newcommand{\Wcal} {{\cal W}}


% GRAS / Boldsymbol
\newcommand{\Alpha} {{\boldsymbol{\alpha}}}
\newcommand{\Beta} {{\boldsymbol{\beta}}}
\newcommand{\xxi} {{\boldsymbol{\xi}}}
\newcommand{\Eta} {{\boldsymbol{\eta}}}
\newcommand{\gammab} {{\boldsymbol{\gamma}}}
\newcommand{\lambdab} {{\boldsymbol{\lambda}}}
\newcommand{\Nu} {{\boldsymbol{\nu}}}
\newcommand{\Varphi} {{\boldsymbol{\varphi}}}
\newcommand{\Rho}{\boldsymbol{\rho}}
\newcommand{\Phib} {{\boldsymbol{\Phi}}}
\newcommand{\Psib} {{\boldsymbol{\Psi}}}
\newcommand{\thetab} {{\boldsymbol{\theta}}}

%Mathbf minuscule
\newcommand{\aab} {{\mathbf{a}}}
\newcommand{\bb} {{\mathbf{b}}}
\newcommand{\cc} {{\mathbf{c}}}
\newcommand{\ddb} {{\mathbf{d}}}
\newcommand{\ee} {{\mathbf{e}}}
\newcommand{\eej} {{\mathbf{e}_j}}
\newcommand{\ff} {{\mathbf{f}}}
\newcommand{\ggb} {{\mathbf{g}}}
\newcommand{\hh} {{\mathbf{h}}}
\newcommand{\jj} {{\mathbf{j}}}
\newcommand{\mm} {{\mathbf{m}}}
\newcommand{\mmh} {\hat{\mathbf{m}}}
\newcommand{\nn} {{\mathbf{n}}}
\newcommand{\pp} {{\mathbf{p}}}
\newcommand{\pph} {\hat{\mathbf{p}}}
\newcommand{\qq} {{\mathbf{q}}}
\newcommand{\qqh} {\hat{\mathbf{q}}}
\newcommand{\rr}{{\mathbf r}}
\newcommand{\ssb} {{\mathbf{s}}}
\newcommand{\uu} {{\mathbf{u}}}
\newcommand{\xx} {{\mathbf{x}}}
\newcommand{\xxh} {{\hat{\mathbf{x}}}}
\newcommand{\yy} {{\mathbf{y}}}
\newcommand{\yyh} {{\mathbf{y}}}
\newcommand{\zz} {{\mathbf{z}}}


%Mathbf Majuscule
% Lettre + Lettre
% ex : \BB
% OU
% Lettre + Lettre + bf
% ex : \AAbf

\newcommand{\Abf} {{\mathbf{A}}}
\newcommand{\BB}{{\mathbf{B}}}
\newcommand{\CC} {{\mathbf{C}}}
\newcommand{\DD} {{\mathbf{D}}}
\newcommand{\EE} {{\mathbf{E}}}
\newcommand{\FF} {{\mathbf{F}}}
\newcommand{\HH} {{\mathbf{H}}}
\newcommand{\II} {{\mathbf{I}}}
\newcommand{\ID} {{\mathbf{\rm{I}}}}
\newcommand{\JJ} {{\mathbf{J}}}
\newcommand{\LL} {{\mathbf{L}}}
\newcommand{\MM} {{\mathbf{M}}}
\newcommand{\NN} {{\mathbf{N}}}
\newcommand{\OO} {{\mathbf{O}}}
\newcommand{\PP} {{\mathbf{P}}}
\newcommand{\QQ} {{\mathbf{Q}}}
\newcommand{\RR} {{\mathbf{R}}}
\newcommand{\UU} {{\mathbf{U}}}
\newcommand{\TT} {{\mathbf{T}}}
\newcommand{\VV} {{\mathbf{V}}}
\newcommand{\XX} {{\mathbf{X}}}
\newcommand{\YY} {{\mathbf{Y}}}

%Boldsymbol Majuscules
% Lettre + Lettre + b
% ex : \AAb

\newcommand{\AAb} {{\boldsymbol{A}}}
\newcommand{\BBb}{{\boldsymbol{B}}}
\newcommand{\CCb} {{\boldsymbol{C}}}
\newcommand{\DDb} {{\boldsymbol{D}}}
\newcommand{\EEb} {{\boldsymbol{E}}}
\newcommand{\FFb} {{\boldsymbol{F}}}
\newcommand{\HHb} {{\boldsymbol{H}}}
\newcommand{\IIb} {{\boldsymbol{I}}}
\newcommand{\IDb} {{\boldsymbol{\rm{I}}}}
\newcommand{\JJb} {{\boldsymbol{J}}}
\newcommand{\LLb} {{\boldsymbol{L}}}
\newcommand{\MMb} {{\boldsymbol{M}}}
\newcommand{\OOb} {{\boldsymbol{O}}}
\newcommand{\PPb} {{\boldsymbol{P}}}
\newcommand{\RRb} {{\boldsymbol{R}}}
\newcommand{\UUb} {{\boldsymbol{U}}}
\newcommand{\TTb} {{\boldsymbol{T}}}
\newcommand{\VVb} {{\boldsymbol{V}}}

%Boldsymbol Majuscules + cal
\newcommand{\MMcal} {{\boldsymbol{\Mcal}}}
\newcommand{\PPcal} {{\boldsymbol{\Pcal}}}


% Tilde + Majuscule
\newcommand{\Ot}{\widetilde{O}}

% Tilde + Majuscule + Gras/boldsymbol
\newcommand{\BBt}{\widetilde{\BB}}
\newcommand{\CCt}{\widetilde{\CC}}
\newcommand{\FFt}{\widetilde{\FF}}
\newcommand{\OOt} {\widetilde{{\mathbf{O}}}}
\newcommand{\UUt} {\widetilde{{\mathbf{U}}}}


% Tilde + minuscule
\newcommand{\bt}{\widetilde{b}}
\newcommand{\dt}{\widetilde{d}}
\newcommand{\ft}{\widetilde{f}}
\newcommand{\rt}{\widetilde{r}}
\newcommand{\rhot}{\widetilde{\rho}}
\newcommand{\mut}{\widetilde{\mu}}
\newcommand{\psit}{\widetilde{\psi}}


% Tilde + minuscule + boldsymbol
\newcommand{\Rhot}{\widetilde{\Rho}}
\newcommand{\lambdabt}{\widetilde{\lambdab}}
\newcommand{\aabt}{\widetilde{\aab}}
\newcommand{\bbt}{\widetilde{\bb}}
\newcommand{\fft}{\widetilde{\ff}}
\newcommand{\rrt}{\widetilde{\rr}}
\newcommand{\ssbt}{\widetilde{\ssb}}
\newcommand{\yyt}{\widetilde{\yy}}
\newcommand{\Psit}{\widetilde{\Psi}}
\newcommand{\xxit}{\widetilde{\xxi}}
\newcommand{\Phit}{\widetilde{\Phi}}

%Tilde + mathbb
\newcommand{\Abt}{\widetilde{\mathbb{A}}}
\newcommand{\Cbt}{\widetilde{\mathbb{C}}}
\newcommand{\Fbt}{\widetilde{\mathbb{F}}}
\newcommand{\Hbt}{\widetilde{\mathbb{H}}}
\newcommand{\Ibt}{\widetilde{\mathbb{I}}}
\newcommand{\Jbt}{\widetilde{\mathbb{J}}}
\newcommand{\Lbt}{\widetilde{\mathbb{L}}}
\newcommand{\Sbt}{\widetilde{\mathbb{S}}}
\newcommand{\Tbt}{\widetilde{\mathbb{T}}}


%Hat + mathbb
\newcommand{\Lbh}{\widehat{\mathbb{L}}}
\newcommand{\Jbh}{\widehat{\mathbb{J}}}
\newcommand{\Hbh}{\widehat{\mathbb{H}}}
\newcommand{\Sbh}{\widehat{\mathbb{S}}}

\newcommand{\etah}{\widehat{\eta}}

%Rm (droit) + minuscule
\newcommand{\n}{{\rm n}}
\newcommand{\s}{{\rm s}}
\newcommand{\x}{{\rm x}}
\newcommand{\y}{{\rm y}}
\newcommand{\dd}{{\rm d}}

% \rm (droit) + majuscule
\newcommand{\F}{{\rm F}}



%Hat + majucule (+ possible boldsymbol)

\newcommand{\FFh}{\widehat{\FF}}

%Hat + minusucle  (+ possible boldsymbol)

\newcommand{\fh}{\widehat{f}}
\newcommand{\ffh}{\widetilde{\ff}}

\newcommand{\rhoh}{\widehat{\rho}}
\newcommand{\Rhoh}{\widehat{\Rho}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Divers
\newcommand{\e} {{\varepsilon}}


\newcommand{\et}      {{\rm e}}
\newcommand{\dy}{{\dd y}}
\newcommand{\dx}{{\dd \xx}}
\newcommand{\dm}   {{\frac{1}{2}}}
\newcommand{\FORALL} {{\hbox{$\hskip 11mm \forall \;$}}}
\newcommand{\kasten} {{\hfill \rule{2.5mm}{2.5mm}}}
\newcommand{\rarrow} {{\,\rightarrow\,}}

\newcommand{\eme} {^{me}}


% ESPACES FONCTIONNELS
\newcommand{\ldo}     {L^2(\Omega)}
\newcommand{\ldg}     {L^2(\Gamma)}
\newcommand{\ldm}     {L^2(\p\Mscr)}
\newcommand{\hun}     {H^1(\Omega)}
\newcommand{\hunz}     {H^1_0(\Omega)}
\newcommand{\hde}     {H^2(\Omega)}
\newcommand{\hm}     {H^m(\Omega)}
\newcommand{\hdemi}     {H^{1/2}(\Gamma)}
\newcommand{\hmdemi}     {H^{-1/2}(\Gamma)}

%\renewcommand{\Re}   {{\rm Re\hskip 0.6mm}}
%\renewcommand{\Im}   {{\rm Im\hskip 0.6mm}}

\newcommand{\dsp}{\displaystyle}
\newcommand{\ima}{\Im{\rm m}\,}
\newcommand{\ree}{\Re{\rm e}\,}
%\newcommand{\ke}{{\rm Ker}}


% INTEGRALES  
\newcommand{\into} {\dsp \int_\Omega}
\newcommand{\intoz} {\dsp \int_{\Omega_0}}
\newcommand{\intg} {\dsp \int_\Gamma}
\newcommand{\intgd} {\dsp \int_{\Gamma^\delta}}
\newcommand{\intgp} {\dsp \int_{\Gamma_p}}
\newcommand{\intgq} {\dsp \int_{\Gamma_q}}
\newcommand{\intgdp} {\dsp \int_{\Gamma^\delta_p}}
\newcommand{\intgdq} {\dsp \int_{\Gamma^\delta_q}}



% DIFF
\newcommand{\p} {\partial}
\newcommand{\ddiv} {{\rm  {div}}} 
\newcommand{\dive} {{\rm  {\bf div}}} 
\newcommand{\divb} {{\rm {\bf div}}_\Beta}
\newcommand{\rot} {{\rm  {\bf curl}}}
\newcommand{\rotb} {{\rm {\bf curl}}_\Beta}  
\newcommand{\dnu} {\dsp \frac{\partial u}{\partial n}}
\newcommand{\dnui} {\dsp \frac{\partial u_I}{\partial n}}
\newcommand{\dnud} {\dsp \frac{\partial u_D}{\partial n}}
\newcommand{\gu} {\dsp\nabla u}
\newcommand{\gv} {\dsp\nabla v}
\newcommand{\gwi} {\dsp\nabla w_I}
\newcommand{\gwj} {\dsp\nabla w_J}
\newcommand{\intouv} {\dsp \into u\,  v\, \dx}
\newcommand{\intoduv} {\dsp \into \Delta u \, v\, \dx}
\newcommand{\intoguv} {\dsp \into \nabla u \cdot \nabla v\, \dx}
\newcommand{\intgdnuv} {\dsp \intg \dnu\, v\, \dx}

% GREEN
\newcommand{\dngi} {\dsp \frac{\partial G_I}{\partial n}}
\newcommand{\dngD} {\dsp \frac{\partial G_D}{\partial n}}


%Permet de numŽroter les Žquations par section
%\renewcommand{\theequation}{{\arabic{section}.\arabic{equation}}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\newcommand{\secp}{{\hbox{\hskip -5.9mm.\hskip 4mm}}}
%%%%%%%%%+%%%%%%%%%+%%%%%%%%%+%%%%%%%%%+%%%%%%%%%+%%%%%%%%%+%%%%%%%%%+



%Matrices par bloc :
\newcommand{\Lbpp}{\mathbb{L}^{p,p}}
\newcommand{\Lbpq}{\mathbb{L}^{p,q}}
\newcommand{\Lbppmm}{\mathbb{L}^{p,p}_{m,m}}
\newcommand{\Lbppmn}{\mathbb{L}^{p,p}_{m,n}}
\newcommand{\Lbpqmn}{\mathbb{L}^{p,q}_{m,n}}
\newcommand{\Lbpqnm}{\mathbb{L}^{p,q}_{n,m}}
\newcommand{\Lbqpnm}{\mathbb{L}^{q,p}_{n,m}}

\newcommand{\Mbpp}{\mathbb{M}^{p,p}}
\newcommand{\Mbpq}{\mathbb{M}^{p,q}}
\newcommand{\Mbppmm}{\mathbb{M}^{p,p}_{m,m}}
\newcommand{\Mbppmn}{\mathbb{M}^{p,p}_{m,n}}
\newcommand{\Mbpqmn}{\mathbb{M}^{p,q}_{m,n}}

\newcommand{\Nbpp}{\mathbb{N}^{p,p}}
\newcommand{\Nbpq}{\mathbb{N}^{p,q}}
\newcommand{\Nbppmm}{\mathbb{N}^{p,p}_{m,m}}
\newcommand{\Nbppmn}{\mathbb{N}^{p,p}_{m,n}}
\newcommand{\Nbpqmn}{\mathbb{N}^{p,q}_{m,n}}

\newcommand{\Dbpp}{\mathbb{D}^{p,p}}
\newcommand{\Dbpq}{\mathbb{D}^{p,q}}
\newcommand{\Dbppmm}{\mathbb{D}^{p,p}_{m,m}}
\newcommand{\Dbppmn}{\mathbb{D}^{p,p}_{m,n}}
\newcommand{\Dbpqmn}{\mathbb{D}^{p,q}_{m,n}}

\newcommand{\Abpp}{\mathbb{A}^{p,p}}
\newcommand{\Abpq}{\mathbb{A}^{p,q}}
\newcommand{\Abppmm}{\mathbb{A}^{p,p}_{m,m}}
\newcommand{\Abppmn}{\mathbb{A}^{p,p}_{m,n}}
\newcommand{\Abpqmn}{\mathbb{A}^{p,q}_{m,n}}

\newcommand{\Ibpp}{\mathbb{I}^{p,p}}
\newcommand{\Ibpq}{\mathbb{I}^{p,q}}
\newcommand{\Ibppmm}{\mathbb{I}^{p,p}_{m,m}}
\newcommand{\Ibppmn}{\mathbb{I}^{p,p}_{m,n}}
\newcommand{\Ibpqmn}{\mathbb{I}^{p,q}_{m,n}}


\newcommand{\AbL}{\Ab_L}
\newcommand{\AbppL}{\AbL^{p,p}}
\newcommand{\AbpqL}{\AbL^{p,q}}
\newcommand{\AbR}{\Ab_R}
\newcommand{\AbppR}{\AbR^{p,p}}
\newcommand{\AbpqR}{\AbR^{p,q}}
\newcommand{\AbM}{\Ab_M}
\newcommand{\AbppM}{\AbM^{p,p}}
\newcommand{\AbpqM}{\AbM^{p,q}}

\newcommand{\Tbpp}{\mathbb{T}^{p,p}}
\newcommand{\Tbpq}{\mathbb{T}^{p,q}}
\newcommand{\Tbppmm}{\mathbb{T}^{p,p}_{m,m}}
\newcommand{\Tbppmn}{\mathbb{T}^{p,p}_{m,n}}
\newcommand{\Tbpqmn}{\mathbb{T}^{p,q}_{m,n}}


\newcommand{\Sbpp}{\mathbb{S}^{p,p}}
\newcommand{\Sbpq}{\mathbb{S}^{p,q}}
\newcommand{\Sbppmm}{\mathbb{S}^{p,p}_{m,m}}
\newcommand{\Sbppmn}{\mathbb{S}^{p,p}_{m,n}}
\newcommand{\Sbpqmn}{\mathbb{S}^{p,q}_{m,n}}

\newcommand{\Jbp}{\Jb^{p}}
\newcommand{\Jbq}{\Jb^{q}}
\newcommand{\Hbp}{\Hb^{p}}
\newcommand{\Hbq}{\Hb^{q}}

\newcommand{\dJb}{(\dd\Jb)}
\newcommand{\dJbp}{(\dd\Jb)^{p}}
\newcommand{\dJbq}{(\dd\Jb)^{q}}
\newcommand{\dHbp}{(\dd\Hb)^{p}}
\newcommand{\dHbq}{(\dd\Hb)^{q}}

\newcommand{\Ibp}{\Ib^{p}}
\newcommand{\Ibq}{\Ib^{q}}

% Avec un chapeau
\newcommand{\Lbhpp}{\Lbh^{p,p}}
\newcommand{\Lbhpq}{\Lbh^{p,q}}
\newcommand{\Lbhppmm}{\Lbh^{p,p}_{m,m}}
\newcommand{\Lbhppmn}{\Lbh^{p,p}_{m,n}}
\newcommand{\Lbhpqmn}{\Lbh^{p,q}_{m,n}}

\newcommand{\Sbhpp}{\Sbh^{p,p}}
\newcommand{\Sbhpq}{\Sbh^{p,q}}
\newcommand{\Sbhppmm}{\Sbh^{p,p}_{m,m}}
\newcommand{\Sbhppmn}{\Sbh^{p,p}_{m,n}}
\newcommand{\Sbhpqmn}{\Sbh^{p,q}_{m,n}}

\newcommand{\Jbhp}{\Jbh^{p}}
\newcommand{\Jbhq}{\Jbh^{q}}
\newcommand{\Hbhp}{\Hbh^{p}}


%Avec un tilde

\newcommand{\Lbtpp}{\widetilde{\mathbb{L}}^{p,p}}
\newcommand{\Lbtpq}{\widetilde{\mathbb{L}}^{p,q}}
\newcommand{\Lbtppmm}{\widetilde{\mathbb{L}}^{p,p}_{m,m}}
\newcommand{\Lbtppmn}{\widetilde{\mathbb{L}}^{p,p}_{m,n}}
\newcommand{\Lbtpqmn}{\widetilde{\mathbb{L}}^{p,q}_{m,n}}

\newcommand{\Mbt}{\widetilde{\mathbb{M}}}
\newcommand{\Mbtpp}{\widetilde{\mathbb{M}}^{p,p}}
\newcommand{\Mbtpq}{\widetilde{\mathbb{M}}^{p,q}}
\newcommand{\Mbtppmm}{\widetilde{\mathbb{M}}^{p,p}_{m,m}}
\newcommand{\Mbtppmn}{\widetilde{\mathbb{M}}^{p,p}_{m,n}}
\newcommand{\Mbtpqmn}{\widetilde{\mathbb{M}}^{p,q}_{m,n}}

\newcommand{\Nbt}{\widetilde{\mathbb{N}}}
\newcommand{\Nbtpp}{\widetilde{\mathbb{N}}^{p,p}}
\newcommand{\Nbtpq}{\widetilde{\mathbb{N}}^{p,q}}
\newcommand{\Nbtppmm}{\widetilde{\mathbb{N}}^{p,p}_{m,m}}
\newcommand{\Nbtppmn}{\widetilde{\mathbb{N}}^{p,p}_{m,n}}
\newcommand{\Nbtpqmn}{\widetilde{\mathbb{N}}^{p,q}_{m,n}}

\newcommand{\Dbt}{\widetilde{\mathbb{D}}}
\newcommand{\Dbtpp}{\widetilde{\mathbb{D}}^{p,p}}
\newcommand{\Dbtpq}{\widetilde{\mathbb{D}}^{p,q}}
\newcommand{\Dbtppmm}{\widetilde{\mathbb{D}}^{p,p}_{m,m}}
\newcommand{\Dbtppmn}{\widetilde{\mathbb{D}}^{p,p}_{m,n}}
\newcommand{\Dbtpqmn}{\widetilde{\mathbb{D}}^{p,q}_{m,n}}

\newcommand{\Cbtpp}{\widetilde{\mathbb{C}}^{p,p}}
\newcommand{\Cbtpq}{\widetilde{\mathbb{C}}^{p,q}}
\newcommand{\Cbtppmm}{\widetilde{\mathbb{C}}^{p,p}_{m,m}}
\newcommand{\Cbtppmn}{\widetilde{\mathbb{C}}^{p,p}_{m,n}}
\newcommand{\Cbtpqmn}{\widetilde{\mathbb{C}}^{p,q}_{m,n}}

\newcommand{\Abtpp}{\widetilde{\mathbb{A}}^{p,p}}
\newcommand{\Abtpq}{\widetilde{\mathbb{A}}^{p,q}}
\newcommand{\Abtppmm}{\widetilde{\mathbb{A}}^{p,p}_{m,m}}
\newcommand{\Abtppmn}{\widetilde{\mathbb{A}}^{p,p}_{m,n}}
\newcommand{\Abtpqmn}{\widetilde{\mathbb{A}}^{p,q}_{m,n}}


\newcommand{\Sbtpp}{\Sbt^{p,p}}
\newcommand{\Sbtpq}{\Sbt^{p,q}}
\newcommand{\Sbtppmm}{\Sbt^{p,p}_{m,m}}
\newcommand{\Sbtppmn}{\Sbt^{p,p}_{m,n}}
\newcommand{\Sbtpqmn}{\Sbt^{p,q}_{m,n}}

\newcommand{\Jbtp}{\Jbt^{p}}
\newcommand{\Jbtq}{\Jbt^{q}}
\newcommand{\Hbtp}{\Hbt^{p}}
\newcommand{\Hbtq}{\Hbt^{q}}
\newcommand{\Ibtp}{\Ibt^{p}}
\newcommand{\Ibtq}{\Ibt^{q}}

\newcommand{\dJbt}{(\dd\Jbt)}
\newcommand{\dJbtp}{(\dd\Jbt)^{p}}
\newcommand{\dJbtq}{(\dd\Jbt)^{q}}
\newcommand{\dHbtp}{(\dd\Hbt)^{p}}
\newcommand{\dHbtq}{(\dd\Hbt)^{q}}

\newcommand{\uut}{\widetilde{\uu}}
\newcommand{\duut}{\widetilde{\dd \uu}}

\newcommand{\Btmp}{\widetilde{B}_m^p}

% Systeme tronquŽ
\newcommand{\Atronq}{\widetilde{\Ab}}
\newcommand{\xitronq}{\widetilde{\xxi}}
\newcommand{\gtronq}{\widetilde{\G}}

% Vecteur entre les centres O_p et O_q et entre l'origine O et Op
\newcommand{\bbpq}{{\mathbf{b}_{pq}}}
\newcommand{\bpq}{b_{pq}}
\newcommand{\alphapq}{\alpha_{pq}}

\newcommand{\bp}{b_{p}}
\newcommand{\bbp}{{\mathbf b}_{p}}
\newcommand{\alphap}{\alpha_{p}}

\newcommand{\Oscrp}{{\cal O}_p}
\newcommand{\Oscrq}{{\cal O}_q}

\newcommand{\OOp}{\OO_p}
\newcommand{\OOq}{\OO_q}

\newcommand{\Op}{O_p}
\newcommand{\Oq}{O_q}

\newcommand{\thetap}{\theta_{p}}
\newcommand{\thetaq}{\theta_{q}}

\newcommand{\ap}{a_p}
\newcommand{\aq}{a_q}

% CoordonnŽes polaires centrŽs sur un obstacle

\newcommand{\rp}{r_p}

\newcommand{\rrp}{{\mathbf r}_p}
\newcommand{\rrq}{{\mathbf r}_q}

%coef du second membre (Fourier)
\newcommand{\dmp}{d_m^p}

% Equations intŽgrales

\newcommand{\EFIED}{L}
\newcommand{\MFIED}{\left(\frac{1}{2}I + N \right)}
\newcommand{\MFIEDext}{\left(-\frac{1}{2}I + N \right)}
\newcommand{\CFIED}{\left[(1-\alpha) \MFIED + \alpha \eta L\right]}

\newcommand{\MFIEN}{\left(\frac{1}{2}I + M \right)}
\newcommand{\MFIENext}{\left(-\frac{1}{2}I + M \right)}
\newcommand{\CFIEN}{\left[(1-\alpha)D  + \alpha \eta \MFIEN\right]}

\newcommand{\BWD}{\left[ -\eta L -M + \frac{1}{2}I \right]}

\newcommand{\CFIEDSecond}{- \left[ (1-\alpha) \dn\uinc|_\Gamma + \alpha \eta \uinc|_\Gamma \right]}
\newcommand{\CFIENSecond}{- \left[(1-\alpha) \duincg + \alpha\eta \uincg \right]}


\newcommand{\OpEFIED}{\Ascr_{EFIE}}
\newcommand{\OpMFIED}{\Ascr_{MFIE}}
\newcommand{\OpCFIED}{\Ascr_{CFIE}}
\newcommand{\OpBWD}{\Ascr_{BW}}

%Vecteur blocs et second membre d'Žquations intŽgrales
\newcommand{\df}{{\mathbf{df}}}

\newcommand{\fq}{{\mathbf{f}}^q}
\newcommand{\fnq}{{\mathbf{f}}^q_n}
\newcommand{\fmp}{{\mathbf{f}}_m^p}
\newcommand{\dfmp}{({\mathbf{df}})_m^p}

\newcommand{\duu}{{\mathbf{du}}}

\newcommand{\uup}{{\mathbf{u}}^p}
\newcommand{\duup}{({\mathbf{du}})^p}
\newcommand{\uump}{{\mathbf{u}}_m^p}
\newcommand{\duump}{({\mathbf{du}})_m^p}

\newcommand{\ggp}{{\mathbf{g}}^p}
\newcommand{\ggmp}{{\mathbf{g}}_m^p}

\newcommand{\qqp}{{\mathbf{q}}^p}
\newcommand{\qqmp}{{\mathbf{q}}_m^p}
 
\newcommand{\uuinc}{\boldsymbol{u}^{inc}}
\newcommand{\duuinc}{\boldsymbol{du}^{inc}}

\newcommand{\Fmp}{F_{m}^p}
\newcommand{\FFmp}{\FF_m^p}

\newcommand{\dFmp}{(\dd \F)_m^p}
\newcommand{\dFFmp}{({\mathbf{dF}})_m^p}

\newcommand{\dUUt} {\dd \UUt}
\newcommand{\dUUtp} {(\dd \UUt)^{p}}
\newcommand{\dUUtmp} {(\dd \UUt)_{m}^{p}}

\newcommand{\CCtp}{\CCt^{p}}
\newcommand{\CCtmp}{\CCt^{p}_{m}}

%Ensembles
\newcommand{\Gammap}{\Gamma_p}
\newcommand{\Gammaq}{\Gamma_q}

\newcommand{\Omegap}{\Omega_p}
\newcommand{\Omegaq}{\Omega_q}

\newcommand{\Omegam}{\Omega^-}
\newcommand{\Omegaps}{\Omega^+}
\newcommand{\Omegapm}{\Omega^{\pm}}


\newcommand{\Omegamp}{\Omega_p^-}
\newcommand{\Omegamq}{\Omega_q^-}

%\newcommand{\Omegab}{\Omega_{b}}

\newcommand{\Omegae}{\Omega_{e}}

\newcommand{\OmegaRE}{\Omega_{RE}}




%Hankel
\newcommand{\Hm}{H_m^{(1)}}
\newcommand{\Hmp}{H_m^{(1)'}}
\newcommand{\Hz}{H_0^{(1)}}
\newcommand{\Hminf}{H_{m_\infty}^{(1)}}
\newcommand{\Hmmoins}{H_{-m}^{(1)}}
\newcommand{\Hmn}{H_{m-n}^{(1)}}
\newcommand{\Hnm}{H_{n-m}^{(1)}}

\newcommand{\Hn}{H_n^{(1)}}
\newcommand{\Hnmoins}{H_{-n}^{(1)}}

\newcommand{\hmm}{h_m^{(1)}}
\newcommand{\hmmp}{h_m^{(1)'}}

%Bessel
\newcommand{\Jm}{J_m}
\newcommand{\Jmmoins}{J_{-m}}
\newcommand{\Jmp}{J'_{m}}

\newcommand{\Jmn}{J_{m-n}}
\newcommand{\Jnm}{J_{n-m}}

\newcommand{\Jn}{J_n}
\newcommand{\Jnmoins}{J_{-n}}
\newcommand{\Jnp}{J'_{n}}

% S_{mn}
\newcommand{\Smn}{S_{mn}}
\newcommand{\Snm}{S_{nm}}
\newcommand{\Smnbpq}{S_{mn}(\bbpq)}
\newcommand{\Snmbpq}{S_{nm}(\bbpq)}

% S_{mn} chapeau
\newcommand{\Smnchap}{\widehat{S}_{mn}}
\newcommand{\Snmchap}{\widehat{S}_{nm}}
\newcommand{\Smnbpqchap}{\widehat{S}_{mn}(\bbpq)}
\newcommand{\Snmbpqchap}{\widehat{S}_{nm}(\bbpq)}

%Base de Fourier : phi_{m}^p
\newcommand{\phim}{\varphi_m}
\newcommand{\phimmoins}{\varphi_{-m}}
\newcommand{\phimp}{\varphi_m^p}
\newcommand{\phimq}{\varphi_m^q}

\newcommand{\phin}{\varphi_n}
\newcommand{\phinmoins}{\varphi_{-n}}
\newcommand{\phinp}{\varphi_n^p}
\newcommand{\phinq}{\varphi_n^q}

\newcommand{\phimn}{\varphi_{m-n}}
\newcommand{\phinm}{\varphi_{n-m}}

%DensitŽs dŽcomposŽes
\newcommand{\rhom}{\rho_{m}}
\newcommand{\rhomp}{\rho_{m}^p}

\newcommand{\rhon}{\rho_{n}}
\newcommand{\rhonq}{\rho_{n}^q}

\newcommand{\lambdam}{\lambda_{m}}
\newcommand{\lambdamp}{\lambda_{m}^p}

\newcommand{\lambdan}{\lambda_{n}}
\newcommand{\lambdanq}{\lambda_{n}^q}

\newcommand{\psim}{\psi_{m}}
\newcommand{\psimp}{\psi_{m}^p}

\newcommand{\psin}{\psi_{n}}
\newcommand{\psinq}{\psi_{n}^q}

\newcommand{\xim}{\xi_{m}}
\newcommand{\ximp}{\xi_{m}^p}

\newcommand{\xin}{\xi_{n}}
\newcommand{\xinq}{\xi_{n}^q}


% coef serie de mie
\newcommand{\cmp}{c_{m}^{p}}



%Base de Fourier : Grand Phi

\newcommand{\Phim}{\Phi_m}
\newcommand{\Phimmoins}{\Phi_{-m}}
\newcommand{\Phimp}{\Phi_m^p}
\newcommand{\Phimq}{\Phi_m^q}

\newcommand{\Phin}{\Phi_n}
\newcommand{\Phinmoins}{\Phi_{-n}}
\newcommand{\Phinp}{\Phi_n^p}
\newcommand{\Phinq}{\Phi_n^q}

\newcommand{\Phimn}{\Phi_{m-n}}
\newcommand{\Phinm}{\Phi_{n-m}}

\newcommand{\Phimt}{\Phit_m}
\newcommand{\Phimmoinst}{\Phit_{-m}}
\newcommand{\Phimpt}{\Phit_m^p}
\newcommand{\Phinpt}{\Phit_n^p}
\newcommand{\Phimqt}{\Phit_m^q}

% Theoreme d'addition

\newcommand{\rrun}{\rrt_1}
\newcommand{\run}{\rt_1}

\newcommand{\rrdeux}{\rr_2}
\newcommand{\rdeux}{\rt_2}

\newcommand{\rri}{\rr_i}
\newcommand{\ri}{\rt_i}

\newcommand{\Oun}{\OOt_1}
\newcommand{\Odeux}{\OOt_2}
\newcommand{\OOi}{\OOt_i}

\newcommand{\bbundeux}{\bbt_{12}}
\newcommand{\bundeux}{\bt_{12}}



%Problemes internes
\newcommand{\Pkd}{(P_{k,D})}
\newcommand{\Pkn}{(P_{k,N})}

%R^2, R^3
\newcommand{\Rbb}{\mathbb{R}^2}
\newcommand{\Rbbb}{\mathbb{R}^3}

%DŽrivŽes :
\newcommand{\dn}{\partial_{\mathbf{n}}}
\newcommand{\dny}{\partial_{\mathbf{n}_{\yy}}}
\newcommand{\dnx}{\partial_{\mathbf{n}_{\xx}}}
\newcommand{\dnz}{\partial_{\mathbf{n}_{\zz}}}

%Vecteur
\newcommand{\V}{\overrightarrow}

\newcommand{\Tq}{T_{q}}
\newcommand{\Tpq}{T_{pq}}



\newcommand{\br}{{\mathbf r}}


%onde totale et onde incidentes
\newcommand{\utm}{u_T^-}
\newcommand{\ut}{u_T}

\newcommand{\vt}{v_T}
\newcommand{\vtm}{v_T^{-}}

\newcommand{\uinc}{u^{\textrm{inc}}}
\newcommand{\uincg}{u^{\textrm{inc}}|_\Gamma}

\newcommand{\duincg}{\dn u^{\textrm{inc}}|_\Gamma}

\newcommand{\uialpha}{u_I^{\Alpha}}
\newcommand{\udalpha}{u_D^{\Alpha}}
\newcommand{\ualpha}{u^{\Alpha}}
\newcommand{\utalpha}{u_T^{\Alpha}}

\newcommand{\uibeta}{u_I^{\Beta}}
\newcommand{\udbeta}{u_D^{\Beta}}
\newcommand{\ubeta}{u^{\Beta}}
\newcommand{\utbeta}{u_T^{\Beta}}

%Disque Gershgorin
\newcommand{\Rmp}{R_m^p}
\newcommand{\Cmp}{C_m^p}
\newcommand{\Ai}{\text{Ai}}
\newcommand{\Bi}{\text{Bi}}
\newcommand{\Rmpt}{\widetilde{R_m^p}}

%conditionnement
\newcommand{\cond}{\text{cond}}


% Ker, Imag, Trace, Signe
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Imag}{{\rm Im}}
\newcommand{\tr}{{\rm tr}}
\newcommand{\sg}{{\rm sgn}}

%Comparaison de fonctions, Valeur Absolue,
\newcommand{\GrandO}[1]{O\left( #1 \right)}
\newcommand{\petito}[1]{o\left( #1 \right)}
\newcommand{\Abs}[1]{\left| #1 \right|}
\newcommand{\abs}[1]{| #1 |}

% fonctions

\newcommand{\sinc}{{\rm sinc}}

% Valeurs propres max, min, ... (estimation de conditionnement)

\newcommand{\lmax}{\Val{\lambda_{max}}}
\newcommand{\lmin}{\Val{\lambda_{min}}}

\newcommand{\mumin}{\mu_{min}}
\newcommand{\muminz}{\mumin^0}
\newcommand{\muminu}{\mumin^1}

\newcommand{\mumax}{\mu_{max}}
\newcommand{\mumaxz}{\mumax^0}
\newcommand{\mumaxu}{\mumax^1}

\newcommand{\mumaxapp}{\mumax^{app}}
\newcommand{\muminapp}{\mumin^{app}}

\newcommand{\mumaxh}{\mumax^{h}}
\newcommand{\muminh}{\mumin^{h}}

% pas equivalent et a equivalent

\newcommand{\heq}{h_{\rm eqv}}
\newcommand{\aeq}{a_{\rm eqv}}

% Produit scalaire

\newcommand{\PS}[2]{\left< #1,  #2 \right>}
\newcommand{\PSLS}[2]{\left( #1,  #2 \right)_{L^2(S^2)}}
\newcommand{\PSLM}[2]{\left( #1,  #2 \right)_{L^2(\Mrt)}}
\newcommand{\PSGamma}[2]{\left( #1,  #2 \right)_{L^2(\Gamma)}}
\newcommand{\PSGammap}[2]{\left( #1,  #2 \right)_{L^2(\Gammap)}}
\newcommand{\PSGammah}[2]{\left( #1,  #2 \right)_{L^2(\Gamma_{h})}}
\newcommand{\PSdemi}[2]{\left< #1,  #2 \right>_{H^{-1/2},H^{1/2}}}

% Equations intŽgrales

\newcommand{\vm}{v^-}
\newcommand{\vmp}{v^-_p}
\newcommand{\vmq}{v^-_q}

\newcommand{\Lpp}{L^{p,p}}
\newcommand{\Lpq}{L^{p,q}}
\newcommand{\Lqp}{L^{q,p}}

\newcommand{\Mpp}{M_{pp}}
\newcommand{\Mpq}{M_{pq}}

\newcommand{\Npp}{N_{pp}}
\newcommand{\Npq}{N_{pq}}

\newcommand{\Dpp}{D_{pp}}
\newcommand{\Dpq}{D_{pq}}


%OpŽrateurs intŽgraux volumiques et divers commandes liŽes aux Žquations intŽgrales
\newcommand{\dwg}{\dn \omega |_\Gamma}
\newcommand{\wg}{\omega |_\Gamma}

\newcommand{\dnwun}{\dn \omega |_{\Gamma_1}}
\newcommand{\wgun}{\omega |_{\Gamma_1}}

\newcommand{\Lop}{\Lscr}
\newcommand{\Mop}{\Mscr}


\newcommand{\Mopp}{\Mop_p}
\newcommand{\Lopp}{\Lop_p}

\newcommand{\Mopq}{\Mop_q}
\newcommand{\Lopq}{\Lop_q}

\newcommand{\Iop}{\Iscr}

\newcommand{\rhop}{\rho_p}
\newcommand{\rhoq}{\rho_q}

\newcommand{\mup}{\mu_p}
\newcommand{\muq}{\mu_q}

\newcommand{\psip}{\psi_p}
\newcommand{\psiq}{\psi_q}


\newcommand{\rhoe}{\rho^*}

\newcommand{\lambdap}{\lambda_p}
\newcommand{\lambdaq}{\lambda_q}

% Vecteurs particulier (utilisŽs dans l'estimation du conditionnement)
\newcommand{\XXpm}{\XX^{(p,m)}}
\newcommand{\XXpmm}{\XX^{(p,-m)}}
\newcommand{\XXpmq}{\XX^{(p,m)}_{q}}
\newcommand{\XXpmqn}{X^{(p,m)}_{q,n}}
\newcommand{\XXpmp}{\XX^{(p,m)}_{p}}
\newcommand{\XXpmpn}{X^{(p,m)}_{p,n}}

\newcommand{\YYpm}{\YY^{(p,m)}}
\newcommand{\YYpmq}{\YY^{(p,m)}_{q}}
\newcommand{\YYpmqn}{Y^{(p,m)}_{q,n}}
\newcommand{\YYpmp}{\YY^{(p,m)}_{p}}
\newcommand{\YYpmpn}{Y^{(p,m)}_{p,n}}

\newcommand{\YYt}{\widetilde{\YY}}
\newcommand{\Yt}{\widetilde{Y}}

\newcommand{\YYtpm}{\YYt^{(p,m)}}
\newcommand{\YYtpmq}{\YYt^{(p,m)}_{q}}
\newcommand{\YYtpmqn}{\Yt^{(p,m)}_{q,n}}
\newcommand{\YYtpmp}{\YY^{(p,m)}_{p}}
\newcommand{\YYtpmpn}{\Yt^{(p,m)}_{p,n}}

%Matrice limite
%matrice L^0
\newcommand{\Lbz}{\Lb^0}
\newcommand{\Lbzpp}{(\mathbb{L}^0)^{p,p}}
\newcommand{\Lbzqq}{(\mathbb{L}^0)^{q,q}}
\newcommand{\Lbzpq}{(\mathbb{L}^0)^{p,q}}
\newcommand{\Lbzppmm}{(\mathbb{L}^0)^{p,p}_{m,m}}
\newcommand{\Lbzppmn}{(\mathbb{L}^0)^{p,p}_{m,n}}
\newcommand{\Lbzpqmn}{(\mathbb{L}^0)^{p,q}_{m,n}}
\newcommand{\Lbzpqnm}{(\mathbb{L}^0)^{p,q}_{n,m}}
\newcommand{\Lbzqpnm}{(\mathbb{L}^0)^{q,p}_{n,m}}

\newcommand{\muz}{(\mu^0)}
\newcommand{\muzmp}{(\mu^0)_{m}^p}

\newcommand{\Lbu}{\Lb^1}
\newcommand{\Lbupp}{\mathbb{L}^1_{p,p}}
\newcommand{\Lbuqq}{\mathbb{L}^1_{q,q}}
\newcommand{\Lbupq}{\mathbb{L}^1_{p,q}}

\newcommand{\muu}{(\mu^1)}
\newcommand{\muump}{(\mu^1)_{m}^p}

%% Chap 1 : RT

% COORDONNEES - Points

\newcommand{\xxz}{\xx_{0}}
\newcommand{\xu}{\x_{1}}
\newcommand{\xuu}{\x_{2}}
\newcommand{\xuuu}{\x_{3}}

\newcommand{\xxc}{\xx^{c}}
\newcommand{\xcu}{\x^{c}_{1}}
\newcommand{\xcuu}{\x^{c}_{2}}
\newcommand{\xcuuu}{\x^{c}_{3}}

\newcommand{\yu}{\y_{1}}
\newcommand{\yuu}{\y_{2}}
\newcommand{\yuuu}{\y_{3}}

\newcommand{\su}{\s_{1}}
\newcommand{\suu}{\s_{2}}
\newcommand{\suuu}{\s_{3}}

\newcommand{\aj}{a_{j}}

% Fonction Phi_{y}...
\newcommand{\phiy}{\phi_{\yy}}
\newcommand{\psiy}{\psi_{\yy}}

\newcommand{\phiyt}{\phi_{\yyt}}
\newcommand{\psiyt}{\psi_{\yyt}}


\newcommand{\phiyxu}{\phi_{\yy;\xuuu}}
\newcommand{\psiyxu}{\psi_{\yy;\xuuu}}

\newcommand{\Phiy}{\Phi_{\yy}}
\newcommand{\Psiy}{\Psi_{\yy}}

% Champ rŽtro-propagŽ
\newcommand{\ue}{u_{E}}
\newcommand{\ve}{v_{E}}

\newcommand{\ure}{u_{RE}}
\newcommand{\vre}{v_{RE}}

\newcommand{\uet}{\widetilde{\ue}}
\newcommand{\vet}{\widetilde{\ve}}

\newcommand{\uret}{\widetilde{\ure}}
\newcommand{\vret}{\widetilde{\vre}}

\newcommand{\UE}{\Uscr_{E}}
\newcommand{\URE}{\Uscr_{RE}}

\newcommand{\ureh}{\widehat{u}_{RE}}
\newcommand{\vreh}{\widehat{v}_{RE}}

\newcommand{\Ve}{\Vscr_{E}}
\newcommand{\Vre}{\Vscr_{RE}}

\newcommand{\Ue}{\Uscr_{E}}
\newcommand{\Ure}{\Uscr_{RE}}
%intŽgrale J(k,\yy)
\newcommand{\Jy}{J_{\yy}}
\newcommand{\Jyt}{J_{\yyt}}
\newcommand{\Jky}{J_{\yy}(k)}
\newcommand{\Jkyxu}{J_{\yy}(k,\xuu)}

\newcommand{\Iy}{I_{\yy}}
\newcommand{\Iky}{I_{\yy}(k)}
\newcommand{\Ikyxu}{I_{\yy}(k,\xuu)}

% DROITE

\newcommand{\Dys}{\Dcal_{\yy\ssb}}
\newcommand{\Dyst}{\Dcal_{\yy\ssbt}}

\newcommand{\Dsa}{\Dcal_{\ssb\Abf}}

\newcommand{\Dxs}{\Dcal_{\xx\ssb}}
\newcommand{\Dxps}{\Dcal_{\xx'\ssb}}
\newcommand{\Dxpst}{\Dcal_{\xx'\ssbt}}
\newcommand{\Dxst}{\Dcal_{\xx\ssbt}}

% Pas artificiel, Rayon moyen...
\newcommand{\htt}{\tilde{h}}
\newcommand{\att}{\tilde{a}}

%UnitŽ
\newcommand{\rad}{{\rm rad}}

% Nombre d'onde, longueur d'onde, ...

\newcommand{\kref}{k_{{\rm ref}}}
\newcommand{\lambdaref}{\lambda_{{\rm ref}}}

\newcommand{\kgeo}{k_{{\rm geo}}}
\newcommand{\lambdageo}{\lambda_{{\rm geo}}}

\newcommand{\kdis}{k_{{\rm dis}}}
\newcommand{\lambdadis}{\lambda_{{\rm dis}}}

\newcommand{\kz}{k_{0}}
\newcommand{\lambdaz}{\lambda_{0}}

\newcommand{\kmin}{k_{{\rm min}}}
\newcommand{\lambdamin}{\lambda_{{\rm min}}}

\newcommand{\kmax}{k_{{\rm max}}}
\newcommand{\lambdamax}{\lambda_{{\rm max}}}

% RT champ proche

\newcommand{\Mrt}{\Mcal}
\newcommand{\Mrtc}{\overline{\Mrt}}

\newcommand{\intM}{\int_{\Mrt}}


\newcommand{\deltaM}{\delta_{\Mrt}}

\newcommand{\dmin}{d_{{\rm min}}}
\newcommand{\dsmin}{d_{\ssb,{\rm min}}}
\newcommand{\dMmin}{d_{\Mrt,{\rm min}}}

\newcommand{\amin}{a_{{\rm min}}}
\newcommand{\amax}{a_{{\rm max}}}

\newcommand{\Deltatheo}{\Delta_{{\rm theo}}}

\newcommand{\Deltaobs}{\Delta_{{\rm obs}}^{num}}

\newcommand{\Deltahom}{\Delta_{{\rm hom}}^{num}}

\newcommand{\Deltainh}{\Delta_{{\rm diff}}^{num}}

\newcommand{\Deltahet}{\Delta_{{\rm diff}}^{num}}

% PML : ondes, ensembles, ...

\newcommand{\upml}{u_{\PML}}
\newcommand{\upmlt}{\tilde{u}_{\PML}}


\newcommand{\PML}{{\rm PML}}
\newcommand{\OmegaPML}{\Omega_{\PML}}
\newcommand{\Sigmaint}{\Sigma_{int}}
\newcommand{\SigmaPML}{\Sigma_{\PML}}
\newcommand{\deltaPML}{\delta^{{\rm PML}}}


%TransformŽe de Fourier
\newcommand{\TF}{{\rm TF}}

\newcommand{\UEh}{\widehat{\Uscr}_{E}}
\newcommand{\UREh}{\widehat{\Uscr}_{RE}}

% DiscrŽtisation en k

\newcommand{\deltak}{\delta_{k}}
\newcommand{\Deltak}{\Delta_{k}}
\newcommand{\dk}{d_{k}}
\newcommand{\Nk}{N_{k}}
\newcommand{\hk}{h_{k}}

% Green

\newcommand{\Gk}{G_{k}}

% Chapitre : estimation conditionnement

\newcommand{\ax}{a_{x}}
\newcommand{\ay}{a_{y}}
\newcommand{\axu}{a_{x_{1}}}
\newcommand{\axuu}{a_{x_{2}}}

% Chapitre : preconditionnement

\newcommand{\LA}{L_{A}}
\newcommand{\LB}{L_{B}}
\newcommand{\LBW}{L_{BW}}

\newcommand{\LBWpp}{L_{BW}^{p,p}}
\newcommand{\LBWpq}{L_{BW}^{p,q}}
\newcommand{\LBWqp}{L_{BW}^{q,p}}

\newcommand{\gammaz}{\gamma_{0}}
\newcommand{\gammazp}{\gamma_{0}^{p}}
\newcommand{\gammazm}{\gamma_{0}^{-}}
\newcommand{\gammazps}{\gamma_{0}^{+}}
\newcommand{\gammazpm}{\gamma_{0}^{\pm}}
\newcommand{\gammazpmd}{\gamma_{\ast}^{\pm}}

\newcommand{\gammau}{\gamma_{1}}
\newcommand{\gammaup}{\gamma_{1}^{p}}
\newcommand{\gammaum}{\gamma_{1}^{-}}
\newcommand{\gammaups}{\gamma_{1}^{+}}
\newcommand{\gammaupm}{\gamma_{1}^{\pm}}

\newcommand{\LAsgl}{\widehat{L}_{A}}
\newcommand{\LBsgl}{\widehat{L}_{B}}
\newcommand{\LBWsgl}{\widehat{L}_{BW}}
\newcommand{\Lsgl}{\widehat{L}}


\newcommand{\LApp}{\LA^{p,p}}
\newcommand{\LAqq}{\LA^{q,q}}
\newcommand{\LApq}{\LA^{p,q}}
\newcommand{\LAqp}{\LA^{q,p}}

\newcommand{\LBpp}{\LB^{p,p}}
\newcommand{\LBqq}{\LB^{q,q}}
\newcommand{\LBpq}{\LB^{p,q}}
\newcommand{\LBqp}{\LB^{q,p}}

\newcommand{\Ap}{A_{p}}
\newcommand{\Aq}{A_{q}}

\newcommand{\Bp}{B_{p}}
\newcommand{\Bq}{B_{q}}

\newcommand{\LAuu}{\LA^{1,1}}
\newcommand{\LAud}{\LA^{1,2}}
\newcommand{\LAuM}{\LA^{1,M}}
\newcommand{\LAdu}{\LA^{2,1}}
\newcommand{\LAdd}{\LA^{2,2}}
\newcommand{\LAdM}{\LA^{2,M}}
\newcommand{\LAMu}{\LA^{M,1}}
\newcommand{\LAMd}{\LA^{M,2}}
\newcommand{\LAmM}{\LA^{M-1,M}}
\newcommand{\LAMM}{\LA^{M,M}}


\newcommand{\LAsgluu}{\LAsgl^{1,1}}
\newcommand{\LAsglud}{\LAsgl^{1,2}}
\newcommand{\LAsgluM}{\LAsgl^{1,M}}
\newcommand{\LAsgldu}{\LAsgl^{2,1}}
\newcommand{\LAsgldd}{\LAsgl^{2,2}}
\newcommand{\LAsgldM}{\LAsgl^{2,M}}
\newcommand{\LAsglMu}{\LAsgl^{M,1}}
\newcommand{\LAsglMd}{\LAsgl^{M,2}}
\newcommand{\LAsglmM}{\LAsgl^{M-1,M}}
\newcommand{\LAsglMM}{\LAsgl^{M,M}}


\newcommand{\LBuu}{\LB^{1,1}}
\newcommand{\LBud}{\LB^{1,2}}
\newcommand{\LBuM}{\LB^{1,M}}
\newcommand{\LBdu}{\LB^{2,1}}
\newcommand{\LBdd}{\LB^{2,2}}
\newcommand{\LBdM}{\LB^{2,M}}
\newcommand{\LBMu}{\LB^{M,1}}
\newcommand{\LBMd}{\LB^{M,2}}
\newcommand{\LBmM}{\LB^{M-1,M}}
\newcommand{\LBMM}{\LB^{M,M}}


\newcommand{\LAt}{\widetilde{L}_{A}}
\newcommand{\LBt}{\widetilde{L}_{B}}

\newcommand{\LAtpp}{\LAt^{p,p}}
\newcommand{\LAtqq}{\LAt^{q,q}}
\newcommand{\LAtpq}{\LAt^{p,q}}
\newcommand{\LAtqp}{\LAt^{q,p}}

\newcommand{\Np}{N_{p}}
\newcommand{\Nq}{N_{q}}
\newcommand{\Ntot}{N_{\textrm{tot}}}
\newcommand{\Nref}{N_{ref}}

\newcommand{\Id}{{\rm Id}}

\newcommand{\diag}{{\rm diag}}

\newcommand{\dFF}{\mathbf{\dd F}}
\newcommand{\dFFt}{\mathbf{\dd}\FFt}

\newcommand{\dUU}{\mathbf{\dd \UU}}



\newcommand{\Angle}{{\rm Angle}}

\newcommand{\AEFIED}{\Ab_{\rm EFIE}}
\newcommand{\AMFIED}{\Ab_{\rm MFIE}}
\newcommand{\ACFIED}{\Ab_{\rm CFIE}}
\newcommand{\ABWD}{\Ab_{\rm BW}}
\newcommand{\AMieD}{\Ab_{\rm Mie}}

\newcommand{\AEFIEDt}{\Abt_{\rm EFIE}}
\newcommand{\AMFIEDt}{\Abt_{\rm MFIE}}
\newcommand{\ACFIEDt}{\Abt_{\rm CFIE}}
\newcommand{\ABWDt}{\Abt_{\rm BW}}
\newcommand{\AMieDt}{\Abt_{\rm Mie}}

\newcommand{\MatEFIEDt}{\Lbt}
\newcommand{\MatMFIEDt}{\frac{1}{2}\Ibt + \Nbt}
\newcommand{\MatCFIEDt}{ \alpha\eta\Lbt + (1-\alpha) \left(\frac{1}{2}\Ibt + \Nbt\right)}
\newcommand{\MatBWDt}{-\eta\Lbt - \Mbt + \frac{1}{2}\Ibt}

\newcommand{\MatEFIED}{\Lb}
\newcommand{\MatMFIED}{\frac{1}{2}\Ib + \Nb}
\newcommand{\MatCFIED}{ \alpha\eta\Lb + (1-\alpha) \left(\frac{1}{2}\Ib + \Nb\right)}
\newcommand{\MatBWD}{-\eta\Lb - \Mb + \frac{1}{2}\Ib}

\newcommand{\AEFIEDpp}{\AEFIED^{p,p}}
\newcommand{\AMFIEDpp}{\AMFIED^{p,p}}
\newcommand{\ACFIEDpp}{\ACFIED^{p,p}}
\newcommand{\ABWDpp}{\ABWD^{p,p}}
\newcommand{\AMieDpp}{\AMieD^{p,p}}

\newcommand{\AEFIEDpq}{\AEFIED^{p,q}}
\newcommand{\AMFIEDpq}{\AMFIED^{p,q}}
\newcommand{\ACFIEDpq}{\ACFIED^{p,q}}
\newcommand{\ABWDpq}{\ABWD^{p,q}}
\newcommand{\AMieDpq}{\AMieD^{p,q}}

\newcommand{\AEFIEDtpp}{\AEFIEDt^{p,p}}
\newcommand{\AMFIEDtpp}{\AMFIEDt^{p,p}}
\newcommand{\ACFIEDtpp}{\ACFIEDt^{p,p}}
\newcommand{\ABWDtpp}{\ABWDt^{p,p}}
\newcommand{\AMieDtpp}{\AMieDt^{p,p}}

\newcommand{\AEFIEDtpq}{\AEFIEDt^{p,q}}
\newcommand{\AMFIEDtpq}{\AMFIEDt^{p,q}}
\newcommand{\ACFIEDtpq}{\ACFIEDt^{p,q}}
\newcommand{\ABWDtpq}{\ABWDt^{p,q}}
\newcommand{\AMieDtpq}{\AMieDt^{p,q}}

\newcommand{\AEFIEDdiag}{\widehat{\Ab}_{\rm EFIE}}
\newcommand{\AMFIEDdiag}{\widehat{\Ab}_{\rm MFIE}}
\newcommand{\ACFIEDdiag}{\widehat{\Ab}_{\rm CFIE}}
\newcommand{\ABWDdiag}{\widehat{\Ab}_{\rm BW}}
\newcommand{\AMieDdiag}{\widehat{\Ab}_{\rm Mie}}

\newcommand{\AEFIEDtdiag}{\widehat{\Abt}_{\rm EFIE}}
\newcommand{\AMFIEDtdiag}{\widehat{\Abt}_{\rm MFIE}}
\newcommand{\ACFIEDtdiag}{\widehat{\Abt}_{\rm CFIE}}
\newcommand{\ABWDtdiag}{\widehat{\Abt}_{\rm BW}}
\newcommand{\AMieDtdiag}{\widehat{\Abt}_{\rm Mie}}

\newcommand{\AEFIEDtpre}{\Abt_{\rm EFIE}^{precond}}
\newcommand{\AMFIEDtpre}{\Abt_{\rm MFIE}^{precond}}
\newcommand{\ACFIEDtpre}{\Abt_{\rm CFIE}^{precond}}
\newcommand{\ABWDtpre}{\Abt_{\rm BW}^{precond}}


\newcommand{\Lbdiag}{\Lbh}
\newcommand{\Lbdiagpq}{\Lbh^{p,q}}
\newcommand{\Mbdiag}{\Mbh}
\newcommand{\Nbdiag}{\Nbh}
\newcommand{\Dbdiag}{\Dbh}

\newcommand{\Lbtdiag}{\Lbt^{single}}
\newcommand{\Lbtdiagpq}{(\Lbt^{single})^{p,q}}
\newcommand{\Mbtdiag}{\Mbt^{single}}
\newcommand{\Mbtdiagpq}{(\Mbt^{single})^{p,q}}
\newcommand{\Nbtdiag}{\Nbt^{single}}
\newcommand{\Nbtdiagpq}{(\Nbt^{single})^{p,q}}
\newcommand{\Dbtdiag}{\Dbt^{single}}
\newcommand{\Dbtdiagpq}{(\Dbt^{single})^{p,q}}


%base de fourier
\newcommand{\BF}{\Bscr}

% champ lointain et ser

\newcommand{\aLop}{a_{\Lop}}
\newcommand{\aMop}{a_{\Mop}}
\newcommand{\dB}{{\rm dB}}

% solveur

\newcommand{\GMRES}{{\rm GMRES}}

%%%%% Article siam

\newcommand{\Fop} {{\rm{\bf{F}}}}
\newcommand{\Rop} {{\rm{\bf{R}}}}
\newcommand{\Top} {{\rm{\bf{T}}}}

%Ondes incidentes, diffractes, totales...
\newcommand{\EIalphaf} {{\EE_I^{\Alpha,\ff}}}
\newcommand{\HIalphaf} {{\HH_I^{\Alpha,\ff}}}

\newcommand{\Ealphaf} {{\EE^{\Alpha,\ff}}}
\newcommand{\Halphaf} {{\HH^{\Alpha,\ff}}}

\newcommand{\ETalphaf} {{\EE_T^{\Alpha,\ff}}}
\newcommand{\HTalphaf} {{\HH_T^{\Alpha,\ff}}}

\newcommand{\EIf} {{\EE_I^{\ff}}}
\newcommand{\HIf} {{\HH_I^{\ff}}}

\newcommand{\Ef} {{\EE^{\ff}}}
\newcommand{\Hf} {{\HH^{\ff}}}

% RT

\newcommand{\Ccalz}{\Ccal_0}
\newcommand{\Ccalu}{\Ccal_1}