(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evaluate ∫f(z)dz around the unit circle where f(z) is given by the following:

a) [itex]\frac{e^{z}}{z^{3}}[/itex]

b) [itex]\frac{1}{z^{2}sinz}[/itex]

c) [itex] tanh(z) [/itex]

d) [itex]\frac{1}{cos2z}[/itex]

e) [itex]e^{\frac{1}{z}}[/itex]

2. Relevant equations

This is the chapter on Laurent Series, so I'm pretty sure:

[itex]C_{n}=\frac{1}{2πi}\oint\frac{f(z)}{(z-z_{0})^{n+1}}dz[/itex]

Is importnat

3. The attempt at a solution

My teacher has also given the hint to 'isolate the singulatiry' and expand the remaining function. For example: [itex]\frac{1}{z^{2}sinz}=\frac{1}{z^{3}}\frac{z}{sinz}[/itex]

Then I expand [itex]\frac{z}{sinz}=\frac{z}{\sum\frac{(-1)^{j}}{(2j+1)!}z^{2j+1}}[/itex]

My problem is that I'm not sure what to do from here. I'm having trouble reciprocating the sum. I want to use the binomial theorem but am not sure how to apply it for an infinite sum. I also don't know what to do once I get it in summation notation.

Could I use uniform convergence to swap summation and integration and note that the only contribution to the sum is the integral with [itex]\frac{1}{z}[/itex] by Cauchy's integral formula??

[itex]f^{(k)}(z_{0})=\frac{k!}{2πi}\oint\frac{f(z)}{(z-z_{0})^{n+1}}dz[/itex]

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# Homework Help: Laurent Series-Finding Contour Integrals

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