Let $\gamma$ be a piece wise smooth closed curve .show that $∫z^mdz=0$

Question

Let $\gamma$ be a piece wise smooth closed curve .show that $∫z^mdz=0$ where $m=0,1,2...$

since $\gamma$ is closed curve and $z^m$ is analytic so by cauchy theorem

$∫z^mdz=0$ is i am right?

Answer 1

The equality is only correct if $m\neq -1.$ In that case $F(z)=\frac{z^{m+1}}{m+1}$ has $f$ as its derivative. So then \begin{equation} \int_\gamma f(z) dz=\int_0^1F'(\gamma(t))\gamma'(t) dt=F(\gamma(1))-F(\gamma(0))=0, \end{equation} where the last inequality holds because $\gamma$ is closed.