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June 7th, 2019, 12:25 PM  #1 
Newbie Joined: Jun 2019 From: Moscow Posts: 14 Thanks: 0  Restore analyticity
Please help Restore the analyticity of f (z) by u(x, y) when f (0) = 2 Last edited by Lasqa; June 7th, 2019 at 12:37 PM. 
June 7th, 2019, 12:28 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,763 Thanks: 697 
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: 2,194 Thanks: 897 Math Focus: Wibbly wobbly timeywimey 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: 619 Thanks: 391 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: 2,194 Thanks: 897 Math Focus: Wibbly wobbly timeywimey stuff. 
The CauchyRiemann 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: 20,746 Thanks: 2133 
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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analyticity, restore 
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