- #1

Poopsilon

- 294

- 1

## Homework Statement

Let [itex]f(z)[/itex] be a complex function analytic everywhere except at [itex]a[/itex] where it has a singularity. Prove that the function [itex]f(z) - \frac{b_{-1}}{z-a}[/itex] has a primitive in a punctured neighborhood of [itex]a[/itex]. Where [itex]b_{-1}[/itex] is the coeffecient of the n=-1 term in the Laurent expansion of [itex]f(z)[/itex].

## Homework Equations

Well I know Laurent series are compactly convergent. I also know path independence of an arbitrary line integral on a punctured neighborhood of [itex]a[/itex] would imply a primitive, so would having the integral over all closed curves equaling zero.

## The Attempt at a Solution

What I'd really like to do is just expand f(z) into its Laurent series, subtract off the [itex]\frac{b_{-1}}{z-a}[/itex] term, and then push the integral operator through the infinite series and integrate term by term. But I think I would need uniform convergence for that, and not just uniform convergence on compact subsets.

I feel like if I had a deeper understanding of uniform convergence I might see why only having compact convergence would be sufficient, but I don't. Beyond that I'm stumped. Help would be much appreciated, thanks.