# An expression in functional analysis

1. May 23, 2010

### Charles49

Are there any theorems concerning this expression
$$\frac{1}{z}f\left(\frac{1}{z}\right)$$. I appreciate posts of any theorems you can think of.

2. May 23, 2010

### Mute

The closest thing I can think of concerns $f(1/z)/z^2$. If you have a complex function f(w) with a pole at infinity, then

$$\mbox{Res}_{w = \infty} f(w) = -\mbox{Res}_{z = 0} \frac{f(\frac{1}{z})}{z^2}$$

3. May 24, 2010

### Charles49

Thanks, this was exactly what I was looking for!

4. May 24, 2010

### Landau

Haha, sometimes a vague question can get the perfect answer immediately :)