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\centerline{{\it  Class 24} {\bf (Wednesday March 25)}}
 
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\item[SUMMARY] Understanding Contour Integration 
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\item[CURRENT READING] Brown \& Curchill pages 123-125
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\item[NEXT READING] Brown \& Curchill pages 125-129
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\underline{Update on {\it Class 22} and {\it Class 23}}\\
After growing comfortable with evaluating contour integrals using the parametrization method (i.e. using $z(t)$) we introduced more higher-level tools such as the Cauchy-Goursat Theorem, the Path Independence Theorem and the Deformation Invariance Theorem.

\underline{\sc Groupwork}\\
Write down, in your own words, a sentence describing each one of the following theorems. You may also want to write down symbols, pictures or even integrals which help you to understand these theorems.\\
{\bf Cauchy-Goursat Theorem}\\
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{\bf Path Independence Theorem}\\
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{\bf Deformation Invariance Theorem}\\
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\underline{\bf Examples}\\
Consider the following integrals of the function ${\displaystyle f(z) = \frac{1}{z^3+z}}$\\
$A = \oint_{C_1} f(z) \, dz$, $B=\oint_{C_2} f(z) \, dz$, $C=\oint_{C_3} f(z) \, dz$ and $D=\oint_{C_4} f(z) \, dz$

The contours $C_1$, $C_2$, $C_3$ and $C_4$ are as sketched below:
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Which of the following equations are true? Give reasons for your answers.

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\item $A = B$?
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\item $B = C$?
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\item $C = D$?
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\item $D = A$?
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\underline{\bf Exercise}\\
Using partial fractions, we can write ${\displaystyle \frac{1}{z^3+z} = \frac{P}{z+i} + \frac{Q}{z-i} + \frac{R}{z}}$

Find $P$, $Q$ and $R$
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Evaluate $A$, $B$, $C$ and $D$
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