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Finding conditions that assure that a holomorphic function i

  1. Feb 21, 2016 #1
    Let H be a real-valued function of two real variables with continuous first partial derivatives. If h(z)=h(x+i y)= u(x,y)+iv(x,y)is holomorphic in a region V, and H(u,v)=0. Find a condition on H under which one can conclude that h is a constant . Of, course H is the zero function can do the job, but i do not want a trivial condition, any clue?
     
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  3. Feb 21, 2016 #2

    Samy_A

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    If H is the zero function, H(u,v)=0 for any choice of functions u,v, and hence for every holomorphic function in V. So H=0 can't be an answer.

    Hint:
    1) What can you tell about ##\frac{dH}{dx}## and ##\frac{dH}{dy}## in V?
    2) What do you know about the first partial derivatives of the real an imaginary part of an holomorphic function?
     
  4. Feb 21, 2016 #3
    ##\frac{dH}{dx}## and ##\frac{dH}{dy}## in V are continuous, how can this help?
     
  5. Feb 21, 2016 #4

    Samy_A

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    You know that H(u,v)=0.
    More precisely, for ##x+iy \in V##, ##H(u(x,y),v(x,y))=0##.
    That should tell you more about the derivatives (when ##x+iy \in V##) than that they are continuous.
    Also, the first partial derivatives of the real an imaginary part of an holomorphic function satisfy a very specific set of equations.
     
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