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Math Help - Show that u is the real part of analytic function

  1. #1
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    Show that u is the real part of analytic function

    Show that u is the real part of analytic function-phpducgnj.png

    Answer

    u(x,y) is the real part of an analytic function if it has zero Laplacian. Note that


    u_x = 2x
    u_xx = 2

    u_y = -2y - 1
    u_yy = -2

    Therefore, the Laplacian of u is

    u_xx + u_yy = 2 - 2 = 0.

    Is the answer correct.....
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    Re: Show that u is the real part of analytic function

    Quote Originally Posted by mathsforumhelp View Post
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    Answer

    u(x,y) is the real part of an analytic function if it has zero Laplacian. Note that


    u_x = 2x
    u_xx = 2

    u_y = -2y - 1
    u_yy = -2

    Therefore, the Laplacian of u is

    u_xx + u_yy = 2 - 2 = 0.

    Is the answer correct.....

    Yes for both questions you need find the Laplacian and see if it is equal to zero. If it is the function is harmonic. Harmonic functions are the real and imagary parts complex functions.
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