### Author Topic: TUT0401 Quiz4  (Read 1083 times)

#### Siyan Chen

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##### TUT0401 Quiz4
« on: February 14, 2020, 10:59:04 AM »
Evaluate the given integral using Cauchy’s Formula or Theorem.

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

First, we can find that $\frac{e^z \ dz}{z(z-3)}$ is not analytic when $z=0$ and $z=3$,

also, $z=3$ is outside the circle $|z|=2$ and $z=0$ is inside the circle $|z|=2$.

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

By Cauchy Formula,  we can get $$f(z)= \frac{e^z}{z-3} , \ and \ z_{0} = 0$$

Therefore, $$\int_{|z|=2} \frac{e^z \ dz}{z(z-3)} = 2 \pi i f(z_0) =2 \pi i \frac{e^0}{0-3}\ = -\frac{2 \pi i}{3}$$