Homework Help: Complex integral

1. Dec 20, 2006

Wiemster

1. The problem statement, all variables and given/known data
$$\oint _{|z+i|=1} \frac{e^z}{1+z^2} dz =?$$

3. The attempt at a solution

I substituted z+i=z' and [itex]z'=e^{i\theta}[/tex] to arrive at

$$e^{-i} \int _0 ^{2 \pi} \frac{e^{e^{i \theta}}}{-ie^{i \theta}-2} d \theta$$

I have no clue how to solve such an integral, any ideas??

(I also did a similar excercise to arrive at the same integral but now [itex]sin(\pi/4 + exp(i \theta))[/tex] in the numerator. Are these kind of integrals analytically solvable??)

Last edited: Dec 20, 2006
2. Dec 20, 2006

TD

Factor the denominator: z²+1 = (z+i)(z-i), then partial fractions.
Do you know Cauchy's integral formula? It states for a inside C:

$$f(a) = {1 \over 2\pi i} \oint_C {f(z) \over z-a}\, dz$$

3. Dec 20, 2006

Wiemster

Thanks a lot! Should have thought of that of course, but now I know I can also make the others, great help!

4. Dec 20, 2006

TD

No problem

5. Dec 20, 2006

Wiemster

Well, maybe I can bother you with one more question? Most of em I can do, but there is this this one with a denominator 1+z^4 which I don't know how to seperate. I tried (z^2+i)(z^2-i) but then I can't seperate these...

Do you maybe have an idea?

6. Dec 20, 2006

TD

You need to find +/- sqrt(i) and +/- sqrt(-i). It factors like this:

$$\left( {z + \frac{{\sqrt 2 + \sqrt 2 i}}{2}} \right)\left( {z + \frac{{\sqrt 2 - \sqrt 2 i}}{2}} \right)\left( {z - \frac{{\sqrt 2 + \sqrt 2 i}}{2}} \right)\left( {z - \frac{{\sqrt 2 - \sqrt 2 i}}{2}} \right)$$

7. Dec 20, 2006

Wiemster

Worked like a charm! Thanks a lot!

8. Dec 20, 2006

TD

You're welcome