Toronto Math Forum
MAT3342020S => MAT334Tests and Quizzes => Quiz 4 => Topic started by: Siyan Chen on February 14, 2020, 10:59:04 AM

Evaluate the given integral using Cauchy’s Formula or Theorem.
$$\int_{z=2} \frac{e^z \ dz}{z(z3)}$$
First, we can find that $\frac{e^z \ dz}{z(z3)}$ 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, $$\int_{z=2} \frac{e^z \ dz}{z(z3)} = \int_{z=2} \frac{ \frac{e^z }{z3}}{z}dz$$
By Cauchy Formula, we can get $$f(z)= \frac{e^z}{z3} , \ and \ z_{0} = 0$$
Therefore, $$\int_{z=2} \frac{e^z \ dz}{z(z3)} = 2 \pi i f(z_0) =2 \pi i \frac{e^0}{03}\ = \frac{2 \pi i}{3}$$