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

Doubt in Partial derivative of complex variables

  1. May 14, 2015 #1
    Today, I had a class on Complex analysis and my professor wrote this on the board :

    The Laplacian satisfies this equation :

    lap.JPG
    where,

    pla.JPG
    So, how did he arrive at that equation?
     
  2. jcsd
  3. May 14, 2015 #2

    ShayanJ

    User Avatar
    Gold Member

    ## \triangle=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}##. Now you should just use chain rule to write derivatives w.r.t. x and y, in terms of derivative w.r.t. ## z ## and ## \bar z ##.
     
  4. May 14, 2015 #3
    x = (z + z¯)/2
    and
    y = (z - z¯)/2i

    How do I solve this further?

    EDIT:Sorry, I don't know how to write the latex code to represent the "Bar" above "z"
     
  5. May 14, 2015 #4
    I guess, I'll have to use Wirtinger derivatives.
     
  6. May 14, 2015 #5
    By the definition $$\frac{\partial}{\partial z}=\frac12\left(\frac{\partial}{\partial x} -i \frac{\partial}{\partial y} \right), \qquad \frac{\partial}{\partial\overline z}=\frac12\left(\frac{\partial}{\partial x} +i \frac{\partial}{\partial y} \right).$$ So if you multiply these two differential operators and use the fact that $$\frac{\partial}{\partial x}\frac{\partial}{\partial y} = \frac{\partial}{\partial y}\frac{\partial}{\partial x}$$ (equality of mixed partial derivatives), you get exactly the Laplacian.
     
  7. May 14, 2015 #6
    I meant ##1/4## of the Laplacian, i.e. $$\frac14\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right).$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook