# Thread: How to apply General Cauchy's Theorem

1. ## How to apply General Cauchy's Theorem

The question was to evaluate the integral of f(z) dz, around C, where C is the unit circle centered at the origin, using the general cauchy's theorem.

f(z) is 1/((2z^2)+1).

I know that f(z) is not analytic at i(sqrt2)/2 and -i(sqrt2)/2 and both points happen to be inside C, so cauchy's theorem can't be applied.

What's my next step?

2. ## Re: How to apply General Cauchy's Theorem

I don't see how you can use Cauchy's theorem here... unless maybe you mean Cauchy's residue theorem.

But that's not necessary either. It's probably easiest to use Cauchy's integral formula. Draw a line through the unit circle such that each pole is inside one of the resulting half-circles. Let $\gamma_1,\gamma_2$ be appropriately-oriented paths corresponding to each half-circle such that $\int_{|z|=1}f(z)dz=\int_{\gamma_1}f(z)dz+\int_{ \gamma_2}f(z)dz$. Then by Cauchy's integral formulae we have

$\int_{|z|=1}f(z)dz=\int_{\gamma_1}\frac{1/2(z-[i/\sqrt{2}])}{z+[i/\sqrt{2}]}+\int_{\gamma_1}\frac{1/2(z+[i/\sqrt{2}])}{z-[i/\sqrt{2}]}$

$=2\pi i\left[\frac{1}{2(-[i/\sqrt{2}]-[i/\sqrt{2}])}+\frac{1}{2([i/\sqrt{2}]+[i/\sqrt{2}])}\right]$

3. ## Re: How to apply General Cauchy's Theorem

I haven't learned the Cauchy Residue Theorem yet.

Thanks, I think I'm starting to get the hang of it now.