- #1

moxy

- 41

- 0

## Homework Statement

Given C is the unit circle, evaluate [tex]\int_C \frac{1}{z^2 + 4} dz[/tex]

## Homework Equations

unit circle: [itex]z = e^{iθ}[/itex]

The problem doesn't specify how many times to go around the unit circle or which way, so I'm going to assume once and counterclockwise.

## The Attempt at a Solution

[tex]z = e^{iθ} \ \ , \ θ \in [0, 2\pi][/tex]

[tex]dz = ie^{iθ}dθ[/tex]

[tex]\int_C \frac{1}{z^2 + 4} dz = \int_0^{2\pi} \frac{1}{(e^{iθ})^2 + 4} ie^{iθ}dθ[/tex]

[tex] = \frac{1}{2} \arctan{\left(\frac{e^{iθ}}{2}\right)} \Big|_0^{2\pi} [/tex]

[tex] = \frac{1}{2} \arctan{\left(\frac{e^{0i}}{2}\right)} - \frac{1}{2} \arctan{\left(\frac{e^{2\pi i}}{2}\right)} [/tex]

[tex] = \frac{1}{2} \arctan{\left(\frac{1}{2}\right)} - \frac{1}{2} \arctan{\left(\frac{1}{2}\right)} = 0[/tex]

I'm pretty confident about it, but I'm always wary of numerical problems that turn out so nicely. Is this correct, or did I royally screw it up?