# Residue & pole of a function

1. Apr 16, 2006

### quasar987

Here I must evaluate

$$\frac{1}{2\pi i}\int_C \frac{f(\zeta)}{w(\zeta)}\frac{w(\zeta)-w(z)}{\zeta -z}d\zeta$$

where f is holomorphic on the hole complex plane, where w(z) is a polynomial of degree n with all of its zeroes distinct (i.e. all n have multiplicity 1), and where C is a closed curve containing all the zeroes of w in its interior.

The solution manual says that the integral equals

$$\sum_{j=1}^n Res(\frac{f(\zeta)}{w(\zeta)}\frac{w(\zeta)-w(z)}{\zeta -z}, \zeta_j)$$

where $\zeta_j$ are the n zeroes of w. But isn't $\zeta = z$ also a pole for the integrand?

2. Apr 17, 2006

### Tide

If the contour excludes $\zeta = z$ then the pole will not contribute to the integral.