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$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$

$$v(x,y) = \int_0^1 (yu{\Tiny x} (sx,sy)-xu{\Tiny y} (sx,sy)) ds$$

Prove that there exists a holomorphic function $u+iv$ in $Ω$

I know the Cauchy–Riemann equations and what is the result of having a harmonic function (Im not able to write with Latex, otherwise i would write it down) I tried to figure out something with this, but im not coming ahead.

malilini
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  • I tried to patch up you $\LaTeX$; hope I got it right. Cheers! – Robert Lewis Nov 04 '18 at 21:49
  • What do you mean you've tried it in the right way? Have you tried to show that $u,v$ satisfy the Cauchy Riemann equations? – Alex R. Nov 04 '18 at 21:51
  • Are you familiar with the fact two Harmonic functions are conjugates if and only if they are the real and imaginary parts respectively of an analytic function? – Brevan Ellefsen Nov 04 '18 at 22:07
  • thanks Robert Lewis :) – malilini Nov 04 '18 at 22:19
  • Alex R, it was just because i cannot write with latex, but now it stands there how it should be. – malilini Nov 04 '18 at 22:20
  • Yes Brevan Ellefsen, I know that. But anyway i dont know what to do now. – malilini Nov 04 '18 at 22:22
  • @malilini sorry, missed your message since you didn't tag my name. At this point there are two ways you can go: use the CR equations or show $u$ and $v$ are Harmonic conjugates by a more direct method (if you know any) – Brevan Ellefsen Nov 05 '18 at 16:45
  • @Brevan Ellefsen I tried it, but i dont really know how to go ahead. I thought about: $$g=u{\Tiny x}- iu{\Tiny y}$$ and then I tried to use the CR equations... But no... I really don't know how to go on.. – malilini Nov 05 '18 at 20:49

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