Cauchy integral formula in simple connected domain

Question

What are the formulas for:

Cauchy integral formula for function in a simply-connected domain

And

Cauchy integral formula for derivatives in simply-connected domain.

Answer 1

The general formula (which I would note can be found using the google) is, for $\gamma$ the boundary of a disk in the simply connected domain, and $a$ inside the disk, and $f$ of course holomorphic on the domain, we get $$ f^{n}(a)=\frac{n!}{2\pi i}\int_{\gamma}\frac{f(z)}{(z-a)^{n+1}}\mathrm dz $$ If you set $n=0$, you get the formula for the function's value at $a$.