Question 5.29: Evaluate ∫C e^z /(z² + p²)² dz where C is the circle |z|= 4.

Schaum’s Outline of Complex Variables

Evaluate $\oint_C \frac{e^z}{\left(z^2+\pi^2\right)^2} d z$ where C is the circle |z|= 4.

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\text { The poles of } \frac{e^z}{\left(z^2+\pi^2\right)^2}=\frac{e^z}{(z-\pi i)^2(z+\pi i)^2} \text { are at } z=\pm \pi i \text { inside } C \text { and are both of order two. }

\text { Residue at } z=\pi i \text { is } \lim _{z \rightarrow \pi i} \frac{1}{1 !} \frac{d}{d z}\left\{(z-\pi i)^2 \frac{e^z}{(z-\pi i)^2(z+\pi i)^2}\right\}=\frac{\pi+i}{4 \pi^3} \text {. }

\text { Residue at } z=-\pi i \text { is } \lim _{z \rightarrow-\pi i} \frac{1}{1 !} \frac{d}{d z}\left\{(z+\pi i)^2 \frac{e^z}{(z-\pi i)^2(z+\pi i)^2}\right\}=\frac{\pi-i}{4 \pi^3} \text {. }

\text { Then } \oint_C \frac{e^z}{\left(z^2+\pi^2\right)^2} d z=2 \pi i \text { (sum of residues) }=2 \pi i\left(\frac{\pi+i}{4 \pi^3}+\frac{\pi-i}{4 \pi^3}\right)=\frac{i}{\pi} \text {. }

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