Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Quiz 4 => Topic started by: Pengyun Li on October 22, 2020, 07:12:31 PM

Question: Evaluate the given integral using Cauchy’s Formula or Theorem: $\int_{z=2}\frac{e^z}{z(z3)}dz$.
Answer:
For $z(z3)=0$, $z=0$ or $z=3$, where only $z=0$ is bounded by $z=2$, thus $z_0=0$.
$\int_{z=2}\frac{e^z}{z(z3)}dz = \int_{z=2}\frac{\frac{e^z}{z3}}{z0}dz$
(By Cauchy's formula)
$= 2\pi i f(z_0)$, where $f(z) = \frac{e^z}{z3}$ is analytic on $\mathbb{C}$,
Thus, $2\pi i f(z_0) = 2\pi i \cdot \frac{e^0}{03} = \frac{2\pi i}{3}$.
Therefore, $\int_{z=2}\frac{e^z}{z(z3)}dz = \frac{2\pi i}{3}$.