My Math Forum Restore analyticity

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June 7th, 2019, 12:25 PM   #1
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Restore analyticity

Restore the analyticity of f (z) by u(x, y) when f (0) = 2
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Last edited by Lasqa; June 7th, 2019 at 12:37 PM.

 June 7th, 2019, 12:28 PM #2 Global Moderator   Joined: May 2007 Posts: 6,821 Thanks: 723 What is the definition of f? You have an expression for v.
 June 7th, 2019, 12:41 PM #3 Newbie   Joined: Jun 2019 From: Moscow Posts: 14 Thanks: 0 to be honest, I myself do not understand, maybe this is just a new variable for which you need to decide.
 June 7th, 2019, 02:10 PM #4 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,266 Thanks: 934 Math Focus: Wibbly wobbly timey-wimey stuff. So, at a guess, we are to find a function u(x, y) such that with v(x, y) we have an analytic function f = u + iv? -Dan
 June 7th, 2019, 03:36 PM #5 Newbie   Joined: Jun 2019 From: Moscow Posts: 14 Thanks: 0 yes
 June 7th, 2019, 04:08 PM #6 Senior Member   Joined: Sep 2016 From: USA Posts: 642 Thanks: 406 Math Focus: Dynamical systems, analytic function theory, numerics What does it mean to "restore" analyticity? This doesn't make any sense.
 June 7th, 2019, 04:22 PM #7 Newbie   Joined: Jun 2019 From: Moscow Posts: 14 Thanks: 0 Find the imaginary part U (x, y) of this function and make up the function F (z) satisfying the initial condition f (0) = 2
 June 7th, 2019, 06:35 PM #8 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,266 Thanks: 934 Math Focus: Wibbly wobbly timey-wimey stuff. The Cauchy-Riemann equations for an analytic function are $\displaystyle \dfrac{ \partial u}{ \partial x} = \dfrac{ \partial v}{ \partial y}$ And $\displaystyle \dfrac{ \partial u}{ \partial y} = - \dfrac{ \partial v}{ \partial x}$ So the next step is to find equations for u(x, y). You know v(x, y) so you can find $\displaystyle \dfrac{ \partial u}{ \partial x}$ and $\displaystyle \dfrac{ \partial u}{ \partial y}$. Then you need to integrate them. Can you finish? -Dan
 June 7th, 2019, 09:43 PM #9 Global Moderator   Joined: Dec 2006 Posts: 20,966 Thanks: 2216 F(z) = (1 + i/2)z² + 2
 June 8th, 2019, 02:03 AM #10 Newbie   Joined: Jun 2019 From: Moscow Posts: 14 Thanks: 0 Oh, I think I understand how to do it. What then should be done after integration? Compare?

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