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 :)

 

[CellCoree]

I am not an expert in functional analysis, but I can provide a general response to this expression.

In functional analysis, the expression \frac{1}{z}f\left(\frac{1}{z}\right) can be interpreted as the composition of two functions: the reciprocal function \frac{1}{z} and the function f\left(\frac{1}{z}\right). This expression can also be written as \frac{f(z)}{z}, which may be a more familiar form.

Some possible theorems related to this expression include:

1. Theorem: If f(z) is a continuous function on a region R in the complex plane, and \frac{1}{z} is holomorphic on R, then \frac{1}{z}f\left(\frac{1}{z}\right) is also holomorphic on R.

2. Theorem: If f(z) is a holomorphic function on a region R in the complex plane, and \frac{1}{z} is meromorphic on R, then \frac{1}{z}f\left(\frac{1}{z}\right) is also meromorphic on R.

3. Theorem: If f(z) is an entire function (holomorphic on the entire complex plane), then \frac{1}{z}f\left(\frac{1}{z}\right) is also entire.

In addition, the expression \frac{1}{z}f\left(\frac{1}{z}\right) can also be used in the study of conformal mappings, where it can help to transform a region in the complex plane into another region, preserving angles and shapes.

Overall, the expression \frac{1}{z}f\left(\frac{1}{z}\right) is a useful tool in functional analysis and has many applications in complex analysis and other fields of mathematics.