# Complex Analysis: Integrating rational functions

## Homework Statement

Hi all.

My question has to do with integrating rational functions over the unit circle. My example is taken from here (page 2-3): http://www.maths.mq.edu.au/~wchen/lnicafolder/ica11.pdf

We wish to integrate the following

$$\int_0^{2\pi } {\frac{{d\theta }}{{a + \cos \theta }}}.$$

According to the .pdf, we integrate along a unit circle, and we define $z = e^{i\theta}$. When rewriting the integrating using z, we find that the integrand has to poles: One inside the unit circle and one outside. When we use the residue theorem, we only use the pole inside the unit circle.

My question: How are we even allowed just to say: "We choose only to integrate along a unit circle, and thus we only look at poles inside this circle"? If I claim that we should integrate along a circle big enough to include all poles, then who can say argument against my claim?

I hope you understand me.

Best regards,
Niles.

Related Calculus and Beyond Homework Help News on Phys.org
Landau
The reasoning is not "we must integrate along a unit circle, therefore we substitute z=e^(it)", but "let's substitute z=e^(it), then - since t ranges from 0 to 2*pi - we are integrating along a unit circle".
If you want to substitue z=Re^(it), so that you integrate along a non-unit circle, you could try that, but the question is whether that will work. The substitution z=e^(it) is easy to work with.

If you want to substitue z=Re^(it), so that you integrate along a non-unit circle, you could try that, but the question is whether that will work. The substitution z=e^(it) is easy to work with.
I have not yet seen a proof of that substituting z = exp(it) works as well, only that it apparently makes things easy.

Landau