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Is there exists a non-constant real valued function in $D$ which is analytic in $D$?

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  • $\begingroup$ What is $D$? The disk? Some region? Other? $\endgroup$ Oct 29, 2014 at 7:20
  • $\begingroup$ $D$ is a region in $\mathbb C$.@AdamHughes $\endgroup$
    – David
    Oct 29, 2014 at 8:36
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    $\begingroup$ Then the open mapping theorem doesn't allow this. $\endgroup$ Oct 29, 2014 at 8:50
  • $\begingroup$ so???@AdamHughes $\endgroup$
    – David
    Oct 29, 2014 at 8:59

2 Answers 2

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No. For instance, from Cauchy-Riemann equations.

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  • $\begingroup$ So u mean that every analytic real valued function in a region is constant. right??????@orangeskid $\endgroup$
    – David
    Oct 29, 2014 at 8:37
  • $\begingroup$ @David: Yep $\ \ \ \ \ \ \ \ $ $\endgroup$
    – orangeskid
    Oct 29, 2014 at 9:18
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Actually there's a theorem that says that if $\mathbb{D}$ is a region of $\mathbb{C}$ and $f$ is holomorphic in this region, then they're equivalent:

  1. The real part of $f$ is constant
  2. The imaginary part of $f$ is constant
  3. $f$ is constant
  4. $\overline{f}$ is holomorphic
  5. $|f|$ is constant

So as you can see if $f$ is a non-constant real valued function then it can't be holomorphic.

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