Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Harmonic Functions, conjugates and the Hilbert Transform

  1. Apr 26, 2012 #1

    I am currently confused about something I've run across in the literature.

    Given that
    [itex] \nabla^2\phi = \phi_{xx}+\phi_{zz} = 0 [/itex] for [itex] z\in (-\infty, 0] [/itex]


    [itex] \phi_z = \frac{\partial}{\partial x} |A|^2 [/itex] at z=0.

    for [itex] A= a(x)e^{i \theta(x)} [/itex].

    The author claims that

    [itex] \phi_x = A_xA^*-AA^*_x [/itex] at z=0

    and where A* represents the complex conjugate.

    The author then claims a more general formula for [itex] \phi_x [/itex] can be found in terms of the Hilbert Transform.

    I do not understand how the author finds the expression for [itex] \left.\phi_x\right|_{z=0} [/itex]. Also, although I'm vaguely aware that Hilbert Transforms can be used to find Harmonic conjugates, I don't see how that can be exploited in this case.

    Any suggestions are appreciated!


  2. jcsd
  3. May 12, 2012 #2
    What is this publication?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook