Numerical evaluation of a complex integral

Question

I have to evaluate numerically $f(z)$ via the Cauchy representation (so via a complex integral), in other words, I have to calculare $f(z)$ performing a complex integral:

$\dfrac{1}{2\pi i}\displaystyle\oint_\Gamma \dfrac{f(t)}{t-z}dt$

I have found some matlab algoritms that performs this calculations via a mean, I would like to understand where does this procedure derive from.

Could someone explain me this derivation or give me any reference?

Answer 1

If $f(t)$ is a holomorfic function and $\Gamma$ represents a closed integration path that contains $z$, then you can use Cauchy's integral formula

$\displaystyle f(z) = \frac{1}{2\pi i} \oint_\Gamma dt\,\frac{f(t)}{t-z}$.

EDIT:

You have changed the question. To find a derivation, wikipedia gives you a proof for a small circular circuit. You can use Cauchy's integral theorem to understand that any circuit that encloses $z$ must yield the same result.