Thread: Complex Analysis

1. Complex Analysis

Hee I have not yet a good idea how to start on this one.

Let $\displaystyle f(z)=\frac{p(z)}{q(z)}$ a rational function with $\displaystyle \text{deg}(q)\geq \text{deg}(p)+2$.

Show that the sum of the residues of f is zero.

Some guiding hints will be very much appreciated.

2. Originally Posted by Dinkydoe
Hee I have not yet a good idea how to start on this one.

Let $\displaystyle f(z)=\frac{p(z)}{q(z)}$ a rational function with $\displaystyle \text{deg}(q)\geq \text{deg}(p)+2$.

Show that the sum of the residues of f is zero.

Some guiding hints will be very much appreciated.
Integrate f round a circle centred at the origin, with really large radius R. On the circle, $\displaystyle |f(z)| < \text{const.}/R^2$, so the absolute value of the integral will be of order 1/R. As R goes to infinity, the contour contains all the singularities, and the integral ( $\displaystyle = 2\pi i\times$(sum of residues)) goes to 0.

3. Ah thanks,that explains why the condition deg(p)<=deg(q)+2 is necessary