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Let $f(z)$ be a complex and continuous function on $[0,1]$. We'll define $g(z) = \int_0^1 f(t)e^{tz} \, dt $. Prove that $g(z)$ is an entire function.

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    Have you tried the Cauchy-Riemann equations?2011-11-14
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    Morera's theorem seems to do it pretty quickly. See my answer below.2011-11-14
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    You may differentiate under the integral sign.2011-11-14
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    @Christian Maybe a _proof_ that one may differentiate under the integral sign is just about what is being asked for here.2011-11-14

2 Answers 2