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I have given a function $f$ which is continuous on $\Bbb C$ and analytic on the imaginary axis and I have to prove that $f$ is analytic in $\Bbb C$, hence , $f$ is entire.

Would anybody please help me solve this problem? Thank you!

Tota
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  • Given the wide variety of interpretations, it'd be good to update your question on what you mean by analytic on the imaginary axis. I know you've done so in a comment, but it's better to have it in the question itself – gist076923 Mar 29 '23 at 12:49

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This is not true: consider $f(z)=\min(\Re z,1)+i\Im z$

  • this is a theorem, hence it must be true. Would you please be kind and explain why do you think this is not true(proof)? – Tota Apr 06 '20 at 01:30
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    @tea I don't know what you think you have read, but the function I have written is continuous on $\Bbb C$ and holomorphic on ${z\in\Bbb C,:, \Re z<1}$, as you can check for yourself. –  Apr 06 '20 at 01:34
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    @Tea Can you give us the precise statement of that "theorem"? It can't be what you wrote... – Robert Israel Apr 06 '20 at 01:37
  • Let f(z) be continuous on C and analytic on {z| Im(z)=/0}.Prove that f must be analytic in whole complex plane. (I wrote =/ instead of "different from") – Tota Apr 06 '20 at 01:39
  • @Gae.S. , @ Robert Israel , I've found this:https://math.stackexchange.com/questions/1548515/a-continous-function-on-g-complex-analytic-on-g-setminus-s-is-analytic-on , it looks similar , but I'm not sure if it helps... (sorry) – Tota Apr 06 '20 at 01:53
  • Do you see the difference between "$\text{Im}(z) \ne 0$" and "on the imaginary axis"? – Robert Israel Apr 06 '20 at 02:07
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    @Tea ${z\in\Bbb C,:, \Im z\ne0}$ isn't the imaginary axis, which would be ${z\in\Bbb C,:, \Re z=0}$, but the complement of the real line. –  Apr 06 '20 at 02:08
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Hint (for the question you meant to ask): Do you know Morera's theorem?

Robert Israel
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  • Yes I do . And how do you think I can use it? – Tota Apr 06 '20 at 10:38
  • It suffices to show the integral around any triangle is $0$. If the real axis intersects the triangle, look at the parts on either side. – Robert Israel Apr 06 '20 at 13:20
  • would you please be kind and send me the proof of this theorem using the Morera's theorem , because I'm not sure if I understood what you meant by that?Thank you. – Tota Apr 06 '20 at 15:20