complex integration on a closed contour

Question

Sorry for my broken English.

I am a physics undergrad and quite poor at math.

I have to prove that integration of $\int_C (z-a)^n dz=0$. $z$ and $a$ is a complex number and $n$ is an integer except $-1$ and $c$ is an arbitrary closed contour.

I know how to integrate if the contour is a circle. but the problem is, it is an arbitrary contour.

I googled and searched a lot but could not find a solution. If this is duplicate, then I apologize. Please help me!

Answer 1

Just use the fact that $(z-a)^n$ has a primitive: $F(z)=\frac{(z-a)^{n+1}}{n+1}$. So, if the domain of your closed loop $\gamma$ is $[a,b]$,$$\int_\gamma(z-a)^n\,\mathrm dz=F\bigl(\gamma(b)\bigr)-F\bigl(\gamma(a)\bigr)=0.$$

Answer 2

If $ n \in \mathbb Z$ and $n \ne-1$, then the function $(z-a)^n$ has the antiderivative $\frac{(z-a)^{n+1}}{n+1}$.

Can you proceed ?