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Math Help - Solving Analytic complex variable function. (Cauchy Riemann)?

  1. #1
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    Question Solving Analytic complex variable function. (Cauchy Riemann)?

    Hello,
    Can someone please show me how to answer these 2 questions:

    1) Given that f(z) = u(x,y) + iv(x,y) is analytic, and u - v = (x-y)(x^2 + 4xy + y^2), determine the u(x,y) and v(x,y)

    2) For f(z) = u(x,y) + iv(x,y) which is analytic, and its real part is u(x,y) = x^2 + Ay^2.
    a) Determine the constant A in u(x,y).
    b) Given f(0) = 0, determine the function f (z)

    Thanks a load in advance. Please, I am really struggling with these problems.
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  2. #2
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    Because f(z) is analytic we know that u_x  = v_y \,\& \,u_y  =  - v_x.
    Now skipping the tedium in doing the partials, from the given \left\{ \begin{gathered}<br />
  u_x  - v_x  = 3x^2  + 6xy - 3y^2  \hfill \\   u_y  - v_y  = 3x^2  - 6xy - 3y^2  \hfill \\ \end{gathered}  \right.

    From those to we can get 2u_y  = 6x^2  - 6y^2 now solve for u.
    Repeat this process and solve for v and adjust the answers.
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