1. ## 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.

2. 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 $z = 3e^{i\theta}, dz = 3ie^{i\theta}d\theta$.

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.

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

$\int_{c} \big[\frac{1}{z} + \frac{1}{z^3}\big]dz = 2i\pi.$

3. 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?

4. 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