Author Topic: Quiz-5101-D  (Read 346 times)

Pengyun Li

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Quiz-5101-D
« on: October 22, 2020, 07:12:31 PM »
Question: Evaluate the given integral using Cauchy’s Formula or Theorem: $\int_{|z|=2}\frac{e^z}{z(z-3)}dz$.

Answer:

For $z(z-3)=0$, $z=0$ or $z=3$, where only $z=0$ is bounded by $|z|=2$, thus $z_0=0$.

$\int_{|z|=2}\frac{e^z}{z(z-3)}dz = \int_{|z|=2}\frac{\frac{e^z}{z-3}}{z-0}dz$

(By Cauchy's formula)

$= 2\pi i f(z_0)$, where $f(z) = \frac{e^z}{z-3}$ is analytic on $\mathbb{C}$,

Thus, $2\pi i f(z_0) = 2\pi i \cdot \frac{e^0}{0-3} = -\frac{2\pi i}{3}$.

Therefore, $\int_{|z|=2}\frac{e^z}{z(z-3)}dz = -\frac{2\pi i}{3}$.