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Holomorphic functions

  1. Jun 8, 2009 #1
    1. The problem statement, all variables and given/known data
    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 [itex]f(z)=u(x,y)+iv(x,y)[/itex], where we have

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

    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.

    Best regards,
    Niles.
     
  2. jcsd
  3. Jun 8, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

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