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Math Help - Is u harmonic - differentiate u

  1. #1
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    Is u harmonic - differentiate u

    u = z^3 \bar{z}^5 + \bar{z}^3 z^5

    I'm not sure of how to differentiate z z-conjugate. I used z = x + iy and \bar{z} = x - iy and started to differentiate to get u_x. When I get to u_{xx} it's very bloated. Is there a neater way of doing this than my way?

    To check whether u is harmonic or not I need to check if \Delta u = u_{xx} + u_{yy} = 0
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  2. #2
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    Functions involving conjugates are generally not analytic. That means you may just have to do this by brute force (as you've been attempting). I don't see any clever tricks off the top of my head. I'm curious if anyone else comes up with anything quick.
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  3. #3
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    Quote Originally Posted by liquidFuzz View Post
    u = z^3 \bar{z}^5 + \bar{z}^3 z^5

    I'm not sure of how to differentiate z z-conjugate. I used z = x + iy and \bar{z} = x - iy and started to differentiate to get u_x. When I get to u_{xx} it's very bloated. Is there a neater way of doing this than my way?

    To check whether u is harmonic or not I need to check if \Delta u = u_{xx} + u_{yy} = 0
    You can slightly reduce the pain by simplifying before differentiating:

    \begin{aligned}u = z^3 \bar{z}^5 + \bar{z}^3 z^5 &= (z\bar{z})^3(z^2+\bar{z}^2)\\ &= 2(x^2+y^2)^3(x^2-y^2) = 2(x^2+y^2)^2(x^4-y^4) = 2(x^8+2x^6y^2-2x^2y^6-y^8).\end{aligned}

    Now all the tedious work has been done, and the differentiation is easy.
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  4. #4
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    Quote Originally Posted by Opalg View Post
     2(x^8+2x^6y^2-2x^2y^6-y^8)

    Now all the tedious work has been done, and the differentiation is easy.
    Ah... It's like poetry!
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