Understanding Holomorphic Functions: Questions and Solutions

[Niles]

Homework Statement

Hi all.

I have two questions on holomorphic functions in the complex plane.

1) We have shown in class that a holomorphic function f can only depend on z, not z^*, where the asterix denotes complex conjugation.

Today my teacher said that all functions f(z) are holomorphic. He is not correct, is he?

2) I have a holomorphic function $f(z)=u(x,y)+iv(x,y)$, where we have

$$u(x,y)=x^2-y^2+2x \quad \text{and}\quad v(x,y)=2xy+2y.$$

Is there any way that I can find f(z) as a function of z alone? Or is the only method to guess?

Thank you very much in advance.Niles.

 

[Dick]

You don't have to guess. Substitute x=(z+z*)/2 and y=(z-z*)/(2i) and see what you get.