Question

Show that $\frac{1}{2 \pi i} \oint_C \frac{e^{z t}}{z^2+1} d z=\sin t$ if $t>0$ and $C$ is the circle $|z|=3$.

   Show that $\frac{1}{2 \pi i} \oint_C \frac{e^{z t}}{z^2+1} d z=\sin t$ if $t>0$ and $C$ is the circle $|z|=3$.

Schaum's Outlines: Complex Variables

Schaum's Outlines: Complex Variables

Murray Spiegel 1st Edition

Chapter 5, Problem 34

Instant Answer

Step 1

The poles occur where the denominator is zero, i.e., \(z^2 + 1 = 0\). Solving this gives the poles at \(z = i\) and \(z = -i\). Show more…

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Show that $\frac{1}{2 \pi i} \oint_C \frac{e^{z t}}{z^2+1} d z=\sin t$ if $t>0$ and $C$ is the circle $|z|=3$.

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