Using residue theorem, calculate

How do I find the poles of this function?

Printable View

- July 14th 2009, 08:11 AMRichmondResidue Theorem #2
Using residue theorem, calculate

How do I find the poles of this function? - July 14th 2009, 08:25 AMpomp
- July 14th 2009, 08:00 PMRichmond
so

Is that right? - July 14th 2009, 08:07 PMpomp
That is almost right,

What contour would you use to evaluate this integral? Which of the singularities lie within this contour and can you calculate the residues? - July 14th 2009, 08:37 PMRichmond
The four values are

Since the integral limits are and

we take the positive part of the integral and use the positive values

which are

and

So

so the integral will be

=

I think somewhere is wrong. - July 15th 2009, 07:58 AMHallsofIvy
Quite frankly, the fact that you keep writing " " indicates you need to brush up on "complex numbers" before trying contour integrals and residues.

i can be written in "polar form" as . It's fourth roots are given by which gives different result for k= 0, 1, 2, and 3.

With k= 0, a fourth root of i is . With k= 1, another is . With k= 2, . Finally, with k= 3, .

Two of those four roots have positive real part, two negative real part. Two have positive imaginary part, two negative imaginary part. That means you really only need to look at the residues at two poles, depending on how you set up the contour. - July 15th 2009, 08:15 AMOpalg
- July 15th 2009, 08:25 AMRandom Variable

for k = 0, 1, 2, and 3

so the roots are

For the countour integration you only need the two poles in the upper half of complex plane ( and )

- July 15th 2009, 08:51 AMRandom Variable

let

As I mentioned, there are two simple poles in the upper half of the complex plane.

(after simplification)