Results 1 to 3 of 3

Math Help - Harmonic function as a function of z

  1. #1
    Senior Member
    Joined
    Apr 2010
    Posts
    487

    Harmonic function as a function of z

    The question
    For the following harmonic function u : R^2 \to R, find a harmonic conjugate v: R^2 \to R for u and express the analytic function f = u + iv : C \to C as a function of z alone.

    u (x, y) = y^3 - 3yx^2 + 2xy

    My attempt:
    I used Cauchy Riemann equations to find v, and ended up with:

    v = x^3 -3xy^2 - x^2 + C (where C is a constant)

    So the function is f(x, y) = y^3 - 3yx^2 + 2xy + i(x^3 -3xy^2 - x^2 + C) unless I'm mistaken.

    How do I write this in terms of a function of 'z'? I recall my lecturer saying that we set y = 0, solve, then change 'x' to 'z' and this works. Is this OK? I get nervous when I apply something that 'just works' and I have no idea why. Is this a common procedure? Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by Glitch View Post
    How do I write this in terms of a function of 'z'? I recall my lecturer saying that we set y = 0, solve, then change 'x' to 'z' and this works. Is this OK? I get nervous when I apply something that 'just works' and I have no idea why. Is this a common procedure? Thanks.

    That is the Milne-Thompson method, which is a consequence of a well known theorem about analytic continuation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Apr 2010
    Posts
    487
    Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: August 25th 2011, 03:33 PM
  2. Is this function harmonic?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 8th 2009, 04:19 PM
  3. some help about harmonic function
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 28th 2009, 01:10 PM
  4. Harmonic function
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 27th 2009, 11:54 AM
  5. Harmonic Function
    Posted in the Calculus Forum
    Replies: 8
    Last Post: October 19th 2008, 07:39 AM

Search Tags


/mathhelpforum @mathhelpforum