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If we have an entire function $f(z)$, can we apply the Cauchy integration formula for derivative for $f(z)$ and integrate over $\Bbb R$ instead of simple closed curve, i.e.

is this formula is true:

$$f'(a_{0}) = \frac{1}{2\pi i}\int_{\Bbb R}\frac{f(t)}{(t-a_{0})^{2}}dt$$

where $f(t)$ is just $f(z)$ with $z=t\in \Bbb R$.

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    Note: $a_{0}\in \Bbb R$.2011-03-02
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    In other words, prove or disprove that: If $f(z)$ is an entire function, then $$f'(a_{0}) = \frac{1}{2\pi i}\int_{-\infty}^{\infty} \frac{f(t)}{(t-a_{0})^{2}}dt$$ where $f(t)$ is just $f(z)$ with $z=t\in \Bbb R$, and $a_{0}\in \Bbb R$.2011-03-02
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    You can edit the original question rather than make comments to update it. I'm not sure if you knew that.2011-03-02
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    Take f(t) = exp(t). Isn't that a counterexample?2011-03-02
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    You will need some growth behaviour at infinity.2011-04-01

3 Answers 3