Having trouble solving this contour integral. Evaluate the integral e^5z/z^5 where c is the circle |z|=1 traversed in clockwise direction? Any suggestions where to start?
Note that $\displaystyle z=0$ is a pole of order 5. We can apply the residue theorem to this by noting that
$\displaystyle \text{Res}\left(f,0\right)=\frac{1}{4!}\lim_{z\to 0}\frac{\,d^4}{\,dz^4}\left[z^5f\!\left(z\right)\right]=\frac{1}{4!}\lim_{z\to 0}\frac{\,d^4}{\,dz^4}e^{5z}=\frac{1}{4!}\lim_{z\t o0}5^4e^{5z}=\frac{625}{24}$
Thus, $\displaystyle \oint_{\left|z\right|=1}\frac{e^{5z}}{z^5}\,dz=2\p i i\text{Res}\left(f,0\right)=\frac{625\pi i}{12}$
Since we're moving clockwise, it follows that the result will be negative.
Does this make sense?