An expression in functional analysis

[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.

 

[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

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

 

[Charles49]

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

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

Thanks, this was exactly what I was looking for!

 

[Landau]

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