Skip to main content
Contents Index Dark Mode Prev Up Next \(\def\versionnumber{1.54}
\def\JPicScale{0.7}
\def\versionyear{2018}
\def\JPicScale{0.7}
\def\versionmonth{July \versionyear}
\def\JPicScale{0.7}
\def\JPicScale{0.7}
\newcommand{\ix}[1]{#1\index{#1}}
\def\JPicScale{0.7}
\newcommand{\Z}{\mathbb{Z}}
\def\JPicScale{0.7}
\newcommand{\Q}{\mathbb{Q}}
\def\JPicScale{0.6}
\newcommand{\R}{\mathbb{R}}
\def\JPicScale{0.6}
\newcommand{\C}{\mathbb{C}}
\def\s{\sum_{k \geq 0}}
\newcommand{\N}{\mathbb{N}}
\def\sz{\sum_{ k \in \Z }}
\newcommand{\Chat}{\hat\C}
\def\JPicScale{0.7}
\newcommand{\Rhat}{\hat\R}
\def\i{\int_\gg}
\newcommand{\gd}{\delta}
\def\JPicScale{0.7}
\renewcommand{\gg}{\gamma}
\newcommand{\D}{\Delta}
\newcommand{\Dp}{\check{D}}
\DeclareMathOperator{\length}{length}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\fLog}{\mathcal{L}\!og}
\DeclareMathOperator{\fArg}{\mathcal{A}rg}
\DeclareMathOperator{\Log}{Log}
\DeclareMathOperator{\Arg}{Arg}
\let\Im\relax
\let\Re\relax
\DeclareMathOperator{\Im}{Im}
\DeclareMathOperator{\Re}{Re}
\def\sz{\sum_{ k \in \Z }}
\def\s{\sum_{k \geq 0}}
\DeclareMathOperator{\Res}{Res}
\renewcommand{\emptyset}{\varnothing}
\newcommand{\Def}[1]{\textbf{#1}}
\newcommand{\hint}[1]{(\emph{Hint}: #1)}
\newcommand{\histremark}[1]{}
\newcommand{\histremarktwo}[2]{}
\newcommand{\listset}[1]{\left\{#1\right\}}
\newcommand{\makeset}[2]{\listset{#1\colon\,#2}}
\newcommand{\listseq}[1]{\left\langle#1\right\rangle}
\newcommand{\makeseq}[2]{\listseq{#1\colon\,#2}}
\newcommand{\ds}{\displaystyle}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\zbar}{\overline{z}}
\def\o{\overline}
\newcommand{\fderiv}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\sderiv}[2]{\frac{\partial^2 #1}{\partial #2^2}}
\newcommand{\mderiv}[3]{\frac{\partial^2 #1}{\partial #2 \, \partial #3}}
\newcommand{\mat}[1]{\displaystyle\begin{bmatrix} #1 \end{bmatrix}}
\newcommand{\disp}[1]{$\displaystyle#1$}
\renewcommand{\th}{^{ th}}
\newcommand{\boldcdot}{\boldsymbol{\cdot}}
\newcommand{\diff}[1]{{d#1}}
\newcommand{\itref}[1]{\eqref{#1}}
\def\newnotes{
\begin{remarks}
\thenotes.
}
\def\writenote{
\vspace{10pt} \thenotes.
}
\newcommand{\putqed}{\pushQED{\qed}\popQED}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\newcommand{\fillinmath}[1]{\mathchoice{\underline{\displaystyle \phantom{\ \,#1\ \,}}}{\underline{\textstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptscriptstyle\phantom{\ \,#1\ \,}}}}
\)
Chapter 9 Isolated Singularities and the Residue Theorem
\(\frac 1 {r^2}\) has a nasty singularity at \(r=0\text{,}\) but it did not bother Newton—the moon is far enough.
―Edward Witten
\begin{equation*}
\int_{ C[2,3] } \frac{ \exp(z) }{ \sin(z) } \, \diff{z} \,\text{.}
\end{equation*}
We computed this integral in
Example 8.3.10 crawling on hands and knees (but we finally computed it!), by considering various Taylor and Laurent expansions of
\(\exp(z)\) and
\(\frac 1 { \sin(z) }\text{.}\) In this chapter, we develop a calculus for similar integral computations.
