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Math Help - How to show that this function is entire

  1. #1
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    How to show that this function is entire

    Assume that f(z) is entire. Show that e^f(z) is also entire.

    I did the following:

    f(z) = u(x,y) + iv(x,y) = u + iv

    let w(z) = e^(f(z)) = e^(u + iv) = (e^u)(cos(v) + isin(v)) = g(x,y) + ih(x,y)

    but I couldn't figure out what to do... I'd be very glad if you can help me...
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Prove that e^{f(z)} is holomorphic in \mathbb{C} using the sufficient conditions related with the Cauchy Riemann equations.
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  3. #3
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    Quote Originally Posted by FernandoRevilla View Post
    Prove that e^{f(z)} is holomorphic in \mathbb{C} using the sufficient conditions related with the Cauchy Riemann equations.
    Yes, but I couldn't come to a conclusion...
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    By definition, an entire function is a holomorphic function whose domain is the whole complex plane.
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