How to prove that this type of function is analytic

Asked Oct 01 '16 at 10:19

Active Oct 01 '16 at 10:41

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I have the function $$F(z)=\int_{\Bbb R}f(z,t)\ dt$$ $z\mapsto f(z,t)$ is analytic. How can I prove $F$ is analytic?

We assume that the integral exists.

I wanted to show it is derivable by using the theorem of derivation under the $\int$ sign but I don't think I can apply it to functions of a complex variable as I can to functions of real variables.

(In general I guess $F$ will not be analytic, but I want to have some techniques to prove if it is analytic. Which theorem can help me here for example)

edited Oct 01 '16 at 10:41

asked Oct 01 '16 at 10:19

StarBucK

Can we assume that the integral exists ? – Peter Oct 01 '16 at 10:39
1

Yes we assume it exists – StarBucK Oct 01 '16 at 10:40
A related post: http://math.stackexchange.com/questions/439866/holomorphic-parameter-integral – Olivier Oloa Oct 01 '16 at 10:47
Usually Morera's theorem is more convenient than differentiating under the integral. – Daniel Fischer Oct 01 '16 at 10:51
In fact my main question will be : can I use the theorem of derivation under the integral sign to proove it ? (The theorem usually used with functions of the real variable). – StarBucK Oct 01 '16 at 10:59

How to prove that this type of function is analytic

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