Complex Contour Integral

• April 22nd 2010, 07:30 AM
Negativ
Complex Contour Integral
I'm attempting to do a complex contour integral as follows:

Integral along C given as the circle |z|=3 in the CCW (positive) direction. Evaluate integral along C of z^-1 + z^-3 dz directly by parameterization of C.

I parameterized:

z(Ɵ)=3e^iƟ 0<=Ɵ<=2*pi

Since integral of f(z)dz along c is integral from a to b of f(z(Ɵ))z'(Ɵ)dƟ by (reasoning? also, do I need to show this is piecewise continuous or is it already taken care of?)

I then integrate [1/(3e^iƟ)+1/((3e^iƟ)^3)]3ie^iƟ dƟ
=integral from 0 to 2pi i+i/9e^2iƟ dƟ

iƟ + -1/18*i*e^-2iƟ

which when evaluated from 0 to 2pi gives me

[2pi*i - 1/18 * i * 1] - [0- 1/18*i]

= 2 * i * pi for my final answer.

(since e^-2pi*i = e^0= 1)

My question, then, is what logical/reasoning steps have I left out and did I get this right? These integrals not done with the residue method really mess me up.

• April 22nd 2010, 07:46 PM
davismj
Quote:

Originally Posted by Negativ
I'm attempting to do a complex contour integral as follows:

Integral along C given as the circle |z|=3 in the CCW (positive) direction. Evaluate integral along C of z^-1 + z^-3 dz directly by parameterization of C.

I parameterized:

z(Ɵ)=3e^iƟ 0<=Ɵ<=2*pi

Since integral of f(z)dz along c is integral from a to b of f(z(Ɵ))z'(Ɵ)dƟ by (reasoning? also, do I need to show this is piecewise continuous or is it already taken care of?)

From a to b? Not really sure what a to b is, but I assume you mean $\int_{0}^{2\pi}f(e^{i\theta})z'(\theta )d\theta$. You can think of z as a function on theta, or you can just think of it as a change of variables. In other words, if $z = 3e^{i\theta}, dz = 3ie^{i\theta}d\theta$.

Quote:

Originally Posted by Negativ

I then integrate [1/(3e^iƟ)+1/((3e^iƟ)^3)]3ie^iƟ dƟ
=integral from 0 to 2pi i+i/9e^2iƟ dƟ

iƟ + -1/18*i*e^-2iƟ

which when evaluated from 0 to 2pi gives me

[2pi*i - 1/18 * i * 1] - [0- 1/18*i]

= 2 * i * pi for my final answer.

(since e^-2pi*i = e^0= 1)

Okay. I believe you. I couldn't read any of that because of the lack of latex input.

Quote:

Originally Posted by Negativ

My question, then, is what logical/reasoning steps have I left out and did I get this right? These integrals not done with the residue method really mess me up.

Again, really not sure what you wrote above. But, I can assure you that

$\int_{c} \big[\frac{1}{z} + \frac{1}{z^3}\big]dz = 2i\pi.$
• April 23rd 2010, 11:16 AM
Negativ
Thank you very much for the response. What you interpreted from my spastic text was what I was saying.

Edit: How would one type in the more easily readable text?
• April 23rd 2010, 11:58 AM
davismj
Quote:

Originally Posted by Negativ
Thank you very much for the response. What you interpreted from my spastic text was what I was saying.

Edit: How would one type in the more easily readable text?

Write in latex code, and wrap the code in [ MATH ] and [ /MATH ] tags.

LaTeX:Symbols - AoPSWiki