Originally Posted by

**MathSucker** I am trying to work through these simple exercises but it invariably blends together and I get lost. Here is one of them:

Use the Complex FTC and/or Cauchy-Goursat Theorem to show that $\displaystyle \int_\gamma f(z) \,dz\,=0$ , where $\displaystyle \gamma$

is the circle with centre 0 and radius 1, oriented as you prefer, and when:

1. $\displaystyle f(z)=\frac{z}{z-3}$

I'm not sure how to proceed. I know that $\displaystyle \gamma \, : \, e^{it}$ , $\displaystyle t \in [0,2\pi)$

Do I proceed like:

$\displaystyle \int_0^{2\pi} \frac {e^{it}}{e^{it}-3} \, dt \,$

or

$\displaystyle \int_0^{2\pi} \frac {cos(t)+isin(t)}{cos(t)+isin(t)-3} \, dt \,$

Do I just integrate the original equation and forget about polar coordinates?

Surely this is a simple question. Should I be doing something else? Is there any way to avoid tedious integration by parts?