- #1

kreil

Gold Member

- 668

- 68

## Homework Statement

Evaluate the following integral along two different contours, (a) a circle of radius |z|=1/2 centered at the origin, and (b) a circle of radius |z|=3 centered at the origin.

## Homework Equations

[tex]I= \oint \, \frac{dz}{(z^2-1)}[/tex]

## The Attempt at a Solution

I'm not sure that I'm doing this right because I keep getting zero for both integrals (maybe the limits are incorrect?)...

(a) Along this circle, |z|=r=1/2 and [itex]z=re^{i \theta} \, \, dz = i r e^{i \theta} d \theta[/itex] and the integral becomes

[tex]I= \oint \, \frac{dz}{(z^2-1)}= \oint \frac{i r e^{i \theta}d \theta}{r^2-1} = \frac{ir}{r^2-1} \oint_0^{2 \pi}e^{i \theta}d \theta = \frac{ir}{r^2-1} [sin(\theta)-i cos(\theta)]|_0^{2 \pi} = \frac{ir}{r^2-1}(-i+i)=0[/tex]

In part (b) the value of r changes but the integrand stays the same so I get zero again..what am I doing wrong?