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Math Help - Proving that an analytic function on a region is constant.

  1. #1
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    Proving that an analytic function on a region is constant.

    Let f=u+iv be an analytic function defined on a region D. Suppose
    a*u(x,y) + b*v(x,y) = c for all x+iy in D, where a,b,c (in C) are constants, not all 0.
    Prove that f is constant on D.

    My guess is that I'm meant to show df/dx and df/dy = 0? Because it's constant. But I'm not sure how to go about it. I've used the Cauchy-Riemann equations to derive expressions for df/dx and df/dy with u and v, but I'm not really sure what to do with them, nor the terms of a, b and c.
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  2. #2
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    Quote Originally Posted by jimmehhh View Post
    Let f=u+iv be an analytic function defined on a region D. Suppose
    a*u(x,y) + b*v(x,y) = c for all x+iy in D, where a,b,c (in C) are constants, not all 0.
    Prove that f is constant on D.

    My guess is that I'm meant to show df/dx and df/dy = 0? Because it's constant. But I'm not sure how to go about it. I've used the Cauchy-Riemann equations to derive expressions for df/dx and df/dy with u and v, but I'm not really sure what to do with them, nor the terms of a, b and c.


    The given condition is equivalent to f being bounded on D, and since it is entire

    there Liouville Theorem states that it must be constant there.

    Tonio
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