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

Please help
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.
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June 7th, 2019, 12:28 PM   #2
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What is the definition of f? You have an expression for v.
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June 7th, 2019, 12:41 PM   #3
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to be honest, I myself do not understand, maybe this is just a new variable for which you need to decide.
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June 7th, 2019, 02:10 PM   #4
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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
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June 7th, 2019, 03:36 PM   #5
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yes
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June 7th, 2019, 04:08 PM   #6
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What does it mean to "restore" analyticity? This doesn't make any sense.
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June 7th, 2019, 04:22 PM   #7
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Find the imaginary part U (x, y) of this function and make up the function F (z) satisfying the initial condition f (0) = 2
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June 7th, 2019, 06:35 PM   #8
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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
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June 7th, 2019, 09:43 PM   #9
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F(z) = (1 + i/2)z² + 2
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June 8th, 2019, 02:03 AM   #10
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Oh, I think I understand how to do it. What then should be done after integration? Compare?
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