Toronto Math Forum
MAT3342020S => MAT334Tests and Quizzes => Quiz 3 => Topic started by: Jiayue Wu on February 12, 2020, 05:54:31 PM

Question: Evaluate the given integral using Cauchy's Formula or Theorem.
$$\int_{z = 2} \frac{e^z}{z(z3)}dz$$
Answer:
We can find that on the region $z = 2$, $F(z) = \frac{e^z}{z(z3)}$ not continuous at z = 0. Therefore I'll apply Cauchy's Formula.
$$\int_{z = 2} \frac{e^z}{z(z3)}dz = \int_{z = 2} \frac{e^z /(z3)}{z}dz$$
$$\implies f(z) = \frac{e^z}{z3}, f(0) = \frac{1}{3}$$
$$\int_{z = 2} \frac{e^z}{z(z3)}dz = 2\pi i f(0) = \frac{2\pi i}{3}$$