I'm currently reading Freitag&Busam's complex analysis and get confused about the proof of residue theorem. The proof assumes the fact that the integral on a smmoth curve of $(z-a)^n$ vanishes when $n<-1$, but I cannot see why this is true.
Since the function would have a singularity at $a$, all the theorems I know do not work. I've searched in this site and someone suggests using Cauchy's Integral Formula, but in my book this formula is only available when we integrate on a circle. I've gone through this book to look for this result but don't find it anywhere.
Can anyone provide a proof for this?
Best Answer
If $\gamma$ is a closed path and $f$ is differentiable in some neighborhood of $\gamma$ then $\int_{\gamma} f'(z)\, dz=0$. [It doesn't matter if $f$ is not defined at some points inside the path]. You can see this easily using the definition of path integral. Note that $(z-a)^{n} =f'(z)$ where $f(z)=\frac {(z-a)^{n+1}} {n+1}$. Hence the integral is $0$.