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Math Help - Simple question about a holomorphic function

  1. #1
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    Proof: if f holomorphic then f(z)=λz+c

    Hello. I need a bit of help or a tip maybe..

    I am to show that if f is a holomorphic function that is of the form f(x+iy) = u(x) + i*v(y) where u and v are real functions,
    then f(z) = λz+c where λ is a real number and c is a complex one.

    How would I begin to prove this?

    Thanks to everyone in advance.
    Last edited by seijo; September 11th 2012 at 03:39 PM.
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  2. #2
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    Re: Proof: if f holomorphic then f(z)=λz+c

    Quote Originally Posted by seijo View Post
    I am to show that if f is a holomorphic function that is of the form f(x+iy) = u(x) + i*v(x) where u and v are real functions,
    then f(z) = λz+c where λ is a real number and c is a complex one.
    How would I begin to prove this?
    You can't prove it, because it isn't true (unless \lambda = 0).
    Notice that, when considered as f(z) = f(x,y), the assumption is that f(x,y) is holomorphic and \partial{f}/\partial{y} = 0.
    But your f(x+iy) = λ(x+iy)+c varies with y (unless \lambda = 0).
    The thing to prove is that the function must be constant (i.e. it actually is true, but only with \lambda = 0).
    You can prove that by a straighforward application of the C-R equations.
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  3. #3
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    Re: Proof: if f holomorphic then f(z)=λz+c

    I see your point. But I made a typo. It's meant to be f(x+iy) = u(x) + i*v(y) not f(x+iy) = u(x) + i*v(x).

    I suppose in this case it actually does make sense? Would the C-R equations still be the way to go?
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  4. #4
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    Re: Proof: if f holomorphic then f(z)=λz+c

    Yes, it drops straight out:
    Spoiler:


    u_x(x) = v_y(y) \implies u_x(x) = v_y(y) = \lambda for some \lambda \in \mathbb{R}, because u_{xx} = \partial v_y(y) / \partial x = 0.

    Thus u(x) = \lambda x + b_0, v(y) = \lambda y + b_1, so f(z) = f(x+iy) = u(x) + iv(y)

    = (\lambda x + b_0) + i(\lambda y + b_1) = \lambda (x+iy) + (b_0 + ib_1) = \lambda z + c, where c=b_0 + ib_1.

    Thus f(z) = \lambda z + c for some fixed \lambda \in \mathbb{R}, c \in \mathbb{C}.

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