# Math Help - Complex Integrals

1. ## Complex Integrals

"Apply the Cauchy-Goursat theorem to show that $\int_C f(z) \,dz=0$ when the contour C is the circle |z| = 1, in either direction, and $f(z) = \frac{1}{z^2+z+2}$."

After using quadratic formula, $f(z) = \frac{1}{z^2+z+2} = \frac{1}{(z+1+i)(z+1-i)}$.

Tried breaking down to partial fractions:
$\frac{1}{(z+1+i)(z+1-i)} = \frac{A}{z+1+i} + \frac{B}{z+1-i} = \frac{A(z+1-i) + B(z+1+i)}{(z+1+i)(z+1-i)} = \frac{(A+B)z + (A+B) + (-A+B)i}{(z+1+i)(z+1-i)}$

Dead end since we're looking at A+B=0, A+B=1, and -A+B=0. Any suggestions?

2. Where are your poles relative to your contour and the area enclosed by it?

3. 1) You don't have to break it down into partial fractions to see where the poles are!

2) Why are you assuming that $A+B$ and $(B-A)i$ are the real and imaginary parts of the numerator? You can have $A+B=0$ and $(B-A)i = 1$.

4. You need to show that $f(z)$ is holomorphic (Cauchy's integral theorem - Wikipedia, the free encyclopedia)