1. ## Complex integration

Hey guys, quick question. How would I go about showing that this following equation:

$\displaystyle \int_{|z|=3}^{}tan(z).dz$

equals $\displaystyle -4\Pi i$

...I've tried breaking it down to $\displaystyle sin(z) / cos(z)$, then finding the residues using the singularities that occur at $\displaystyle \Pi /2$ and $\displaystyle -\Pi /2$ but that just leads me to an answer of 0.

If anyone could show me how to do this using the residue theorem, I'd really appreciate it. Thanks.

2. Originally Posted by enjam
Hey guys, quick question. How would I go about showing that this following equation:

$\displaystyle \int_{|z|=3}^{}tan(z).dz$

equals $\displaystyle -4\Pi i$

...I've tried breaking it down to $\displaystyle sin(z) / cos(z)$, then finding the residues using the singularities that occur at $\displaystyle \Pi /2$ and $\displaystyle -\Pi /2$ but that just leads me to an answer of 0.
If a function is of the form f(z)/g(z), and the denominator has a simple zero at $\displaystyle z_0$, then the residue there is given by $\displaystyle f(z_0)/g'(z_0)$. In this case, $\displaystyle f(z)/g'(z) = \sin z/(-\sin z) = -1$. So the residue at both poles is –1, giving the integral as $\displaystyle -4\pi i$.