<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt

QUESTION


Let $C$ denote any simple closed contour taken in the
counterclockwise sense and write $$g(w)=\int_C{{z^3+2z\over
(z-w)^3}dz}$$ Show that $g(w)=6\pi iw$ when $w$ is inside $C$ and
$g(w)=0$ when $w$ is outside $C$.


ANSWER


In (*) we want $n=2$, $f(z)=z^3+2z$ and $w=b$. $f^{''}(z)=6z$, so
$g(w)={2\pi i\over 2!}6w =6\pi iw$ if $w$ lies inside $C$. If $w$
lies inside $C$ then $g(w)=0$ by Cauchy's Theorem.


\end{document}
</pre></body></html>