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quasar987

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**[SOLVED] contour integration**

## Homework Statement

I'm really rusty with this. I need to calculate

[tex]2\pi i Res\left(\frac{1}{1+z^4},e^{i\pi/4}\right)[/tex]

## The Attempt at a Solution

Well,

[tex]2\pi iRes\left(\frac{1}{1+z^4},e^{i\pi/4}\right)=\int_{C_{\rho}}\frac{1}{1+z^4}dz[/tex]

where [tex]C_{\rho}[/tex] is a little circle of radius rho centered on [tex]e^{i\pi/4}[/tex], on which there are no singularities. Fine, so let's take [tex]\rho=1/\sqrt{2}[/tex].

Now I need to parametrize C_rho. Take

[tex]\gamma(t)=e^{i\pi/4}+\frac{e^{i2\pi t}}{\sqrt{2}} \ , \ \ \ \ 0\leq t < 1[/tex]

We have

[tex]\frac{d\gamma}{dt}=i\sqrt{2}\pi e^{i2\pi t}[/tex]

So that

[tex]\int_{C_{\rho}}\frac{1}{1+z^4}dz=\int_0^1\frac{i\sqrt{2}\pi e^{i2\pi t}}{1+(e^{i\pi/4}+\frac{1}{\sqrt{2}}e^{i2\pi t})^4}dt[/tex]

Now what?? :grumpy:

I believe the answer is supposed to be [tex]-e^{i\pi/4}/4[/tex]

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