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Math Help - Cauchy Riemann - questions about answer

  1. #1
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    Cauchy Riemann - questions about answer

    The exercise is as follows:
    Harmonic function u is u(x,y) = e^{2x} \sin(ay) + xy + x where a > 0 and z = x + iy in C. Find an analytic function f where f = u + v.

    In the solution it is stated that a = 2 since u and v are harmonic. I don't know how do deduce this.
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  2. #2
    Senior Member roninpro's Avatar
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    Did you try setting up a system of partial differential equations?

    From the Cauchy-Riemann equations, u_x=v_y and u_y=-v_x. We have v_y=1+y+2 e^{2 x} \text{Sin}[a y] and v_x=-x-a e^{2 x} \text{Cos}[a y]. Try integrating and see what you get.
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  3. #3
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    Well , yep I did. But in the solution a is stated right from start. I thought it was determent in some other fashion.
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  4. #4
    Senior Member roninpro's Avatar
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    You actually have to determine a after you take your integrals. You will get a form that forces a=2 in order for the solution to the system to exist.
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