Complex Integral Evaluation: Solving an Integral Using Cauchy's Residue Theorem

[PsychoDash]

Homework Statement

"Evaluate the integral of f(z)=$$\frac{z^5}{1-z^3}$$ around the circle |z|=2 in the positive sense.

Homework Equations

Cauchy's residue theorem?

The Attempt at a Solution

Truthfully, I don't know where to begin. I've done others of these using Cauchy's theorem and either a partial fraction expansion of the function or a series expansion, but this one seems odd. I realize that not posting work is frowned upon, but I'm stuck at the beginning without any work to show. I don't know how to begin.

 

[PsychoDash]

Ok, I've had an idea. If I let x=z³, then the denominator becomes 1-x. Nice, since the series representation of $$\frac{1}{1-x}$$ is well known. Now I'm just not sure what this change of variable does to the numerator. z⁵=z³*z², but I'm not sure how to write z² in terms of z³. My brain fails me.

edit: $$\frac{x^{5/3}}{1-x}$$

But now there are no fractional terms of the series, so no residue. Does that just mean the value of the integral is zero?

 

[Berko]

All 3 poles are inside the contour. Write the denominator as a product of 3 terms. The residues should then be easily evaluated.

 

[PsychoDash]

If all the residues are zero, then the integral gives zero.

In general. the integral is 2pi-i-times the sum of the residues, if I remember correctly.

I realize that. My question is, am I right in saying that there are no residues? I'm thinking this is some form of a "residue at infinity" example, in which case there are new rules, but I don't understand this enough to tell.

 

[Berko]

I edited my remark before seeing your response. Is my new remark more helpful?

 

[PsychoDash]

All 3 poles are inside the contour. Write the denominator as a product of 3 terms. The residues should then be easily evaluated.

3 terms? I see only two. $$1-z^3=(1-z)(z^2+z+1)$$ but the second term is not factorable.

 

[Berko]

Of course it is. Just not with real coefficients. Fundamental Theorem of something or other.

 

[PsychoDash]

Oh bah. And humbug. And whatever other appropriate phrases one uses when they've forgotten something basic.

 

[Berko]

Haha.