Solve for $m=0,1,2,...$ and $n\in\Bbb{Z}$ the integral $\int_{\Gamma_1} z^n(1-z)^mdz$

Question

I need to solve the integral for $m=0,1,2, ...$, and $n\in\Bbb{Z}$

$$\int_{\Gamma_1} z^n(1-z)^m\,dz$$

where $\Gamma_1$ is the circle centered in $0$ with radius $1$.

I'm struggling trying to see this integral in a way I can use Cauchy's Integral Formula or Cauchy's Integral Formula for derivatives, but so far I'm stuck trying to rewrite it. So could anyone give me any hint to point me in the right direction to solve this?

Thanks in advance.

Answer 1

Just use Cauchy's Integral formula: When $n\geq 0$, $z^n (1-z)^m$ is analytic and so the integral is zero. When $n<0$, set $a=0$ and $f(z) = (1-z)^m$, the Cauchy integral formula is

$$f^{(k-1)}(0) = \frac{k!}{2\pi i} \int_\Gamma \frac{f(z)}{z^k} dz.$$ and so putting $k = -n$ gives

$$\int_\Gamma z^n(1-z)^m dz = \frac{2\pi i}{|n|!}\frac{d^{|n|-1}}{dz^{|n|-1}} (1-z)^m\bigg|_{z=0}.$$