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\centerline{{\it  Class 20} {\bf (Wednesday March 4)}}
 
\begin{description}
\item[SUMMARY] Introduction to Contour Integration
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\item[CURRENT READING] Brown \& Curchill, pages 95-104
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\item[NEXT READING] Brown \& Curchill pages 104-122
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\underline{\bf Exercise}\\
First, let's recall how to integrate complex functions of a {\bf real} variable.
Compute the following:
$$(a) \displaystyle{\int_1^2 \frac{-i}{t^2}+(t+2i)^3 \, dt} \hspace{2in}   (b) \displaystyle{\int_0^{\infty} e^{-z^2t} \, dt} \hspace{1in}$$
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{\bf Contour Integration}\\
Integration of a complex function of a {\bf complex} variable is performed on a set of connected points from, say, $z_1$ to $z_2$. It is a {\bf contour integral}.
Given a contour $C$ defined as $z(t)$ for $a \le t \le b$ where $z_1=z(a)$ and
$z_2 = z(b),$
an integral of a complex function of a complex variable $f(z)$ is written as

$${\displaystyle \int_{C} f(z) dz} \qquad \mbox{ or } \qquad  {\displaystyle \int_{z_1}^{z_2} f(z) \ dz}$$

Let $f(z)$ be piecewise continuous on $z(t)$. If $C$ is a {\bf contour} then $z'(t)$ is piecewise continuous on $a \le t \le b$ and we can redefine the integral of $f(z)$ along $C$ as:
$${\displaystyle \int_{C} f(z) \ dz = \int_a^b f[z(t)] z'(t) \ dt}$$

\underline{\bf Examples}\\
Compute ${\displaystyle \int_C \mbox{Re } z \ dz}$ where $C$ is a directed line segment from $z=0$ to $z=1+2i$

ALGORITHM: steps to be taken to complete the process\\
{\bf 1}: write down a parametricization for $C$, $z(t)$\\
{\bf 2}: Convert the integral into an integral in (real) $t$ variables\\
{\bf 3}: Integrate!\\

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\underline{\sc Groupwork}\\
Compute ${\displaystyle \int_C 2 \overline{z}^2 \ dz}$ where $C$ is a directed l
ine segment from $z=2$ to $z=-2$. (Sketch the contour and evaluate the integral.)
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Also ${\displaystyle \int_C 2 \overline{z}^2 \ dz}$, this time using $C$ being
 a circular arc from $z=2$ to $z=-2$. (Sketch the contour and then evaluate the integral.)
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\underline{\bf Exercise}\\
Show that 
$$
{ \displaystyle \int_{C_r} (z-z_0)^n \ dz } = \left\{ 
\begin{array}{cll}
 2 \pi i & n = -1 \\
 & \\
 0 & n \neq -1
\end{array}
\right.
$$
where $n$ is any integer and $C_r$ is a circle of radius $r$ around $z_0$ (what is the equation of such a shape?) traversed {\bf once} in the counter-clockwise direction

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