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Set of Holomorphic functions s.t. |f1|^2 + + |fn|^2 = K

  1. Sep 18, 2011 #1
    1. The problem statement, all variables and given/known data

    Given holomorphic functions f1, ...,fn defined on some open connected set s.t. |f1|2 + ... + |fn|2 = K. K is some constant. Prove all the fj's are constant.

    2. Relevant equations

    Cauchy Riemann:
    fk = u +iv

    ux = vy
    uy = -vx

    3. The attempt at a solution

    For n=1

    |f|^2 = k.
    we have taking derivatives

    u*ux + v*vx = 0
    u*uy + v*vy = 0

    and hence by CR:
    u*ux + v*vx = 0
    v*ux + -u*vx = 0

    if u^2 + v^2 = 0 we are done

    if not the determinant = -(u^2 + v^2) =/= 0. so the only solution is ux = vx = 0.

    (f\bar = complex conjugate of f)

    also f*f\bar = K (again assume K =/=0)

    f\bar = K/f but since f =/= 0 this implies f\bar is holomorphic which is not possible unless f is constant.

    I can neither get the linear algebra nor the solving for f\bar to work with more than one function. If anyone could even give me a hint for n=2 I'd appreciate it. This is not for a course (though I am taking Complex) I got the problem from Narasimhan complex analysis in one variable prob 48.6. Thanks in advance.
     
    Last edited: Sep 18, 2011
  2. jcsd
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