# Integral calculations using Cauchy's Integral formula

## Homework Statement

Calculate the integral

Given $$\int_{C} \frac{e^z}{\pi i - 2z} dz = \int_{C} \frac{e^z}{z-\frac{\pi i}{2}} dz}$$

using Cauchy integral formula.

## Homework Equations

What I know

$$\frac{1}{2\pi i} \int_{C} \frac{f(z)}{z-\zeta} = 2\pi i f(\zeta)$$

## The Attempt at a Solution

This in my little girly mind amounts to

$$2\pi i f(\frac{\pi i}{2}) = \int_{C} \frac{e^z}{z-\frac{\pi}{2}i} dz \Rightarrow \int_{C} \frac{e^z}{z-\frac{\pi}{2}i} dz = 2 \pi \cdot (i) \cdot (i) = -2\pi$$

But people who are wiser than me says to me "Susanne your result is wrong!". Could someone please point out my mistake?

thanks Susanne

Last edited:

Dick