i have put up the question as a jpeg attachment since i dont know how to type all the math symbols.
i would appreciate any help i am quite stuck it is one of the review questions for my monday exam.
Is...
$\displaystyle \displaystyle \int_{\gamma} f(z)\ dz = 2 \pi i \sum_{k} r_{k}$
... where the $\displaystyle r_{k}$ are the residues of the poles inside $\displaystyle \gamma$. In this case the poles are $\displaystyle z=1$ and $\displaystyle z=2$, both inside $\displaystyle \gamma$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Since both of the poles are simple the resides can be calculated as follows
$\displaystyle \displaystyle f(z)=\frac{e^z}{(z-1)(z-2)}$
The residue at $\displaystyle z=1$
$\displaystyle \displaystyle \text{Res}(f,z=1)=\lim_{z\to1}(z-1)\frac{e^z}{(z-1)(z-2)}=\lim_{z \to 1}\frac{e^z}{(z-2)}=-e$
In general an nth order pole's residue can be calculated using the formula from post #4 in this link
http://www.mathhelpforum.com/math-he...tml#post544571