\newcommand{\ds}{\displaystyle}

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\newcommand{\eeqan}{\end{eqnarray*}}


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\newcommand{\diffb}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}} 



		% Rather like blockquote; \subex{***} makes a field
		% slightly indented on both left and right
		% adjust to match "\textwidth"

\newcommand{\subex}{\noindent\hspace*{5mm}\parbox{165mm}}


\newcommand{\gto}{\rightarrow 0}
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	\parbox{#1mm}{#2}
	\hfill
	\parbox{#3mm}{\psfig{figure=HW/figs/#4.eps,width=#3mm}}
	}

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\newcommand{\csch}{\mbox{csch}\,}
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			\kern-.12em\rule[0.05ex]{0.07em}{1.33ex}\kern .55em}}
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\newcommand{\diag}{\mbox{diag}}
\newcommand{\dintbar}[2]{\rule[0.52ex]{0.55em}{0.2ex}\kern-.90em\int_{#1}^{#2}}

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%	\newcommand{\Doteq}{
%	        \setlength{\unitlength}{1 pt}
%	        \begin{picture}(13,8)
%	        \thinlines
%	        \put(3,3.2){\line(1,0){7}}
%	        \put(3,4.8){\line(1,0){7}}
%	        \put(6.5,1.5){\circle*{0.5}}
%	        \put(6.5,6.5){\circle*{0.5}}
%	        \end{picture}
%	        }


\newcommand{\dztoo}{\Delta z \RTO }
\newcommand{\eCx}{{\rm C\kern-.41em\rule[0.06ex]{0.04em}{0.96ex}\kern .42em}}
\newcommand{\eit}{e^{i\theta}}
\newcommand{\SECTIONPROBS}[1]{
		\label{exer:#1}
		\vspace{1.0ex}
		\centerline{\rule[0.6 ex]{4 in}{0.02 in}}
		\vspace{0.0 ex}
		\centerline{{\bf Problems for section \thesectionNoOp}}
		\vspace{1.0 ex}
		\input HW/#1}
\newcommand{\EndTheExample}{
		\vspace{0.9 ex}
		\centerline{\rule[0.6 ex]{4 in}{0.02 in}}
		\vspace{0.9 ex}}
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	\widetilde{\scriptscriptstyle{#1}}}

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	\displaystyle #2 \end{array} \right. } %}
\newcommand{\threechoices}[3]{\left\{ \begin{array}{lcc}
	#1 \\ #2 \\ #3 \end{array} \right. }	%}
\newcommand{\fourchoices}[4]{\left\{ \begin{array}{lcc}
	#1 \\ #2 \\ #3 \\ #4 \end{array} \right. }	%}

\newcommand{\inta}{\int^{\infty}_{-\infty}}
\newcommand{\intbar}[2]{\rule[0.65ex]{0.47em}{0.3ex}\kern-.72em\int_{#1}^{#2}}
\newcommand{\intc}{\int_C}
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\newcommand{\ldztoo}{\lim_{\dztoo}}
\newcommand{\limjti}{\lim_{j\RTI}}
\newcommand{\limnti}{\lim_{n\RTI}}
\newcommand{\limtoinf}{\lim_{n\RTI}}
\newcommand{\limxtoi}{\lim_{x\RTI}}
\newcommand{\limztoi}{\lim_{z\RTI}}
\newcommand{\limzzo}{\lim_{\zzo}}
\newcommand{\limz}[1]{\lim_{z \rightarrow #1}}
\newcommand{\mQED}{ ~~~~~ \mbox{\rule{2.5mm}{2.5mm}}}
\newcommand{\Na}{{\rm I\kern-.23em N}}    % as in $n \in \Na$
\newcommand{\niki}{-\infty < k < \infty}
\newcommand{\nixi}{-\infty < x < \infty}
\newcommand{\nizi}{-\infty < z < \infty}
\newcommand{\nii}[1]{-\infty < #1 < \infty}
\newcommand{\noi}[1]{      0 < #1 < \infty}
\newcommand{\nitpii}{\frac{-1}{2\pi i}}
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\newcommand{\nth}[1]{$#1^{\scriptstyle th}$}
\newcommand{\nst}[1]{$#1^{\scriptstyle st}$}
\newcommand{\ointc}{\oint_C}
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\newcommand{\ooplus}{\psfig{figure=figs/oplus.eps,height=9pt}}
\newcommand{\oominus}{\psfig{figure=figs/ominus.eps,height=9pt}}
\newcommand{\optionmark}{}
\newcommand{\optstar}{${}^{}\!\!\!*$}
\newcommand{\oz}{ \overline{z} }
\newcommand{\Pain}{Painlev\'e}
\newcommand{\point}[1]{\hfill \\ {\bf #1}}
\newcommand{\prodjii}{\prod_{j=1}^{\infty}}
\newcommand{\prodkii}{\prod_{k=1}^{\infty}}
\newcommand{\prodkin}{\prod_{k=1}^n}
\newcommand{\prodnii}{\prod_{n=1}^{\infty}}
\newcommand{\QED}{\hspace*{\fill}\rule{2.5mm}{2.5mm} \vspace*{\baselineskip}\par}
\newcommand{\regR}{\mbox{$\cal R$}}
\newcommand{\Res}{\mbox{Res}\,}
\newcommand{\Rl}{{\rm I\kern-.29em R}}
\newcommand{\RTO}{\rightarrow 0}
\newcommand{\RTOP}{\rightarrow 0^+}
\newcommand{\Rch}{Rouch\'e}
\newcommand{\Sch}{Schr\"odinger}
\newcommand{\sech}{\mbox{sech}\,}
\newcommand{\tinysech}{\mbox{\tiny sech}\,}
\newcommand{\sgn}{\mbox{sgn}}
\newcommand{\simto}{\longrightarrow\kern-1.25em \sim\kern0.55em}

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	\begin{picture}(6,5) \thinlines
	\put(0,0){\framebox(5,5)} \end{picture}}


\newcommand{\starteqabc}{\setcounter{subsubsection}{1}
	\renewcommand{\theequation}
	{\thesectionNoOp.\arabic{equation}\alph{subsubsection}}}
\newcommand{\nexteqabc}{\addtocounter{equation}{-1}
	\addtocounter{subsubsection}{1}}
\newcommand{\stopeqabc}
	{\renewcommand{\theequation}{\thesectionNoOp.\arabic{equation}}}


\newcommand{\starteqspecial}[1]{\renewcommand{\theequation}
	{\thesectionNoOp.\arabic{equation}#1}}
\newcommand{\stopeqspecial}{\renewcommand{\theequation}
	{\thesectionNoOp.\arabic{equation}}}


\newcommand{\startCVTsection}[2]{\renewcommand{\optionmark}{#1}
				\section{#2}\setcounter{equation}{0}}
\newcommand{\startCVTsubsection}[2]{\renewcommand{\suboptionmark}{#1}
					\subsection{#2}}

% \newcommand{\straight}[2]{\left[\begin{array}{c} #1 \\ #2 \end{array}\right]}

\newcommand{\twovec}[2]{\left(\begin{array}{c} #1 \\ #2 \end{array}\right)}
\newcommand{\threevec}[3]{\left(\begin{array}{c} #1 \\ #2 \\ #3 \end{array}\right)}
\newcommand{\twomatrix}[4]{\left(\begin{array}{cc} #1 & #2 \\ #3 & #4 \end{array}\right)}

\newcommand{\suboptionmark}{}
\newcommand{\sumjii}{\sum_{j=1}^{\infty}}
\newcommand{\sumjin}{\sum_{j=1}^n}
\newcommand{\sumjiN}{\sum_{j=1}^N}
\newcommand{\sumjoi}{\sum_{j=0}^{\infty}}
\newcommand{\sumjon}{\sum_{j=0}^n}
\newcommand{\sumjok}{\sum_{j=0}^k}
\newcommand{\sumkii}{\sum_{k=1}^{\infty}}
\newcommand{\sumkin}{\sum_{k=1}^n}
\newcommand{\sumjnii}{\sum_{j=-\infty}^{\infty}}
\newcommand{\sumknii}{\sum_{k=-\infty}^{\infty}}
\newcommand{\sumkoi}{\sum_{k=0}^{\infty}}
\newcommand{\sumkon}{\sum_{k=0}^n}
\newcommand{\sumnii}{\sum_{n=1}^{\infty}}
\newcommand{\sumniN}{\sum_{n=1}^N}
\newcommand{\sumnnii}{\sum_{n=-\infty}^{\infty}}
\newcommand{\sumnoi}{\sum_{n=0}^{\infty}}
\newcommand{\summoi}{\sum_{m=0}^{\infty}}
\newcommand{\summii}{\sum_{m=1}^{\infty}}
\newcommand{\wi}{w_{\infty}}
\newcommand{\wo}{w_0}
\newcommand{\wwo}{w \rightarrow \wo}
\newcommand{\zi}{z_{\infty}}
\newcommand{\zo}{z_0}
\newcommand{\Zz}{{\bf Z}}		%	as in $i\in\Zz$
\newcommand{\zzo}{z \rightarrow \zo}

\newcommand{\s}{\Sch}
\newcommand{\sn}{\mbox{sn}}
\newcommand{\rhoo}{\rho_{_0}}
\newcommand{\p}{\Pain}
\newcommand{\D}{\DB}
\newcommand\nepsilon{\epsilon\!\!/}
\newcommand\intR{\int_{\Re^2}}
\newcommand\Bint{\int\!\!\!\!\int}
\newcommand\B{B\"acklund }
\newcommand\f{Fr\'echet}
\newcommand\F{\uparrow F \downarrow}
\newcommand\Holder{H\"older}
\newcommand\HO{{\tt{H}}\!\!\!\!\bigcirc }
\newcommand\barV{-\!\!\!V}

\newenvironment{makefig}[4]{
	\par	% {height(in.)}{subcaption}{caption}{label}
	\vbox{
	\refstepcounter{figgur}
	$$ {}^{} \rule{0pt}{#1 in} $$
	\centerline{{\small #2}}
	\vspace{0.2 mm}
	\centerline{Figure \thefiggur: ~#3}
	\label{fig:#4}}\par}{}

\newenvironment{makepsfig}[4]{		% #1 = {LABEL/FILE}
	\par				% #2 = {SUBCAPTIONS?}
	\vbox{				% #3 = {CAPTION}
	\refstepcounter{figgur}		% #4 = {,width=9cm?}
	\vspace{6pt}	
	\centerline{\psfig{figure=figs/#1.eps#4}}
	\centerline{{\small #2}}
	\vspace{4 pt}
	\centerline{Figure \thefiggur: ~#3}
	\label{fig:#1}}\par}{}

\renewcommand{\Im}{\mbox{Im}}
\renewcommand{\Re}{\mbox{Re}}
\newcommand{\Imex}{\mbox{\footnotesize Im}}	% for subscripts
\newcommand{\Reex}{\mbox{\footnotesize Re}}	% for subscripts
\newcommand{\ZERO}{\mbox{\bf 0}}


%	Homework Answers
\newcommand{\ansa}[2]{{\bf(#1.)} #2}			% (a) only
\newcommand{\ansm}[2]{\hfill{\bf(#1.)} #2}		% Middle
\newcommand{\ansl}[2]{\hfill{\bf(#1.)} #2 \hfill \\}	% Last in line
\newcommand{\ansf}[2]{\ii{\bf(#1.)} #2}			% First (after newline)
\newcommand{\anso}[2]{\ii{\bf(#1.)} #2 \\}		% Only (after newline)
\newcommand{\ansnl}{\hfill \\}				% force new line
\newcommand{\ansz}{\hfill\iir\\}			% 1 blank filler at end
\newcommand{\anszz}{\hfill\iir\hfill\iir\\}		% 2 blank fillers at end
\newcommand{\anszzz}{\hfill\iir\hfill\iir\hfill\iir\\}	% 3 blank fillers at end