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Math Help - Complex Analysis Proof

  1. #1
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    Question Complex Analysis Proof

    Show that if a function f(z) = u(x,y) + iv(x,y) is entire, that the function conj(f(conj(z))) is entire. (Note that CR stands for Cauchy-Riemann)

    Since f is entire, CR is satisfied:
    so u_x = v_y and u_y = -v_x

    this implies:
    u_x = -(-v_y) and -u_y = -(-v_x)

    this implies:
    CR is satisfied for a function g(z) = u(x,-y) - iv(x,-y)

    but g(z) = conj(f(conj(z)))

    this implies:
    CR is satisfied for conj(f(conj(z)))

    Since f is entire, the partial derivatives are continuous

    this implies:
    conj(f(conj(z))) is entire.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    That is correct.
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