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Math Help - Prove that f is analytic

  1. #1
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    Arrow Prove that f is analytic

    Let \mathcal U\subset\mathbb C be open and consider a function f:[0,1]\times\mathcal U\longrightarrow\mathbb C so that for all t\in[0,1] the function f(t,\cdot):\mathcal U\longrightarrow\mathbb C is analytic. Prove that F(z)=\int_0^1 f(t,z)\,dt is analytic.

    Any ideas? Can I use Morera's Theorem? How?

    Thanks.
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  2. #2
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    Re: Prove that f is analytic

    Quote Originally Posted by Homing View Post
    Let \mathcal U\subset\mathbb C be open and consider a function f:[0,1]\times\mathcal U\longrightarrow\mathbb C so that for all t\in[0,1] the function f(t,\cdot):\mathcal U\longrightarrow\mathbb C is analytic. Prove that F(z)=\int_0^1 f(t,z)\,dt is analytic.

    Any ideas? Can I use Morera's Theorem? How?

    Thanks.
    I think that you need some extra condition for this result to hold. Suppose for example that

    f(t,z) = \begin{cases}z & \text{if }t \text{ is rational,} \\ -z & \text{if }t \text{ is irrational.} \end{cases}

    Then f(t,z) is analytic (as a function of z) for each fixed t\in[0,1]. But it is not (Riemann) integrable as a function of t, except at z=0.

    If you have some sufficiently strong additional condition controlling the integrability of f as a function of t, then you could hope to prove the result by approximating the integral by Riemann sums, and using the fact (corollary of Morera's theorem) that a uniform limit of analytic functions is analytic.
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  3. #3
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    Re: Prove that f is analytic

    Perhaps the continuity of f(t,z) and the connectedness of \mathcal U ?
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