Complex analysis - holomorphic functions Watch

hopelesslylost
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Hi all,
I’m looking for some help with this complex analysis question, part a in particular. Can I get some help starting it? I’m really not sure where to begin. Any help is appreciated.
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DFranklin
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Cauchy Riemann equations should be a good starting point...
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hopelesslylost
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I’ve just done that and I’ve come to the conclusion that f is holomorphic for all x and y. Does that sound right?
(Original post by DFranklin)
Cauchy Riemann equations should be a good starting point...
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DFranklin
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(Original post by hopelesslylost)
I’ve just done that and I’ve come to the conclusion that f is holomorphic for all x and y. Does that sound right?
Seems likely; I've not actually done any calculations, but looking at it I think you can fairly easily write it as a polynomial in z = (x+iy), and so it's differentiable.
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hopelesslylost
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I think I’ve done a) correctly then. Any advice with b)? I’ve got that it was holomorphic on 0, pi/2, etc, but I don’t know how to differentiate it
(Original post by DFranklin)
Seems likely; I've not actually done any calculations, but looking at it I think you can fairly easily write it as a polynomial in z = (x+iy), and so it's differentiable.
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DFranklin
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(Original post by hopelesslylost)
I think I’ve done a) correctly then. Any advice with b)? I’ve got that it was holomorphic on 0, pi/2, etc, but I don’t know how to differentiate it
If you look at the derivation for the Cauchy-Riemann equations, you'll see various expressions for the complex derivative that involve the terms in the Cauchy Riemann equations. (Because the Cauchy-Riemann derivation is basically "find two different expressions that must both equal the complex derivative, and deduce that they must be equal). Just use one of those.

[I could give you a simple "do this" formula, but you shouldn't need me to if you understand the derivation.

I was going to give a longer explanation, but TSR lost my post and I can't face doing it again.

If you need more help, find a derivation of the CR equations online that you understand (or at least is equivalent to your lectures), post a link to it, and I will try to give more help using that as a reference].
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hopelesslylost
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I don't know how to differentiate the solution at all, i'm looking at it, and its just stumping me. I've done the cauchy riemann equations for the first part, to find that its holomorphic on 0, p/2, etc, but I don't how to actually differentiate the function

(Original post by DFranklin)
If you look at the derivation for the Cauchy-Riemann equations, you'll see various expressions for the complex derivative that involve the terms in the Cauchy Riemann equations. (Because the Cauchy-Riemann derivation is basically "find two different expressions that must both equal the complex derivative, and deduce that they must be equal). Just use one of those.

[I could give you a simple "do this" formula, but you shouldn't need me to if you understand the derivation.

I was going to give a longer explanation, but TSR lost my post and I can't face doing it again.

If you need more help, find a derivation of the CR equations online that you understand (or at least is equivalent to your lectures), post a link to it, and I will try to give more help using that as a reference].
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RDKGames
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(Original post by hopelesslylost)
I don't know how to differentiate the solution at all, i'm looking at it, and its just stumping me. I've done the cauchy riemann equations for the first part, to find that its holomorphic on 0, p/2, etc, but I don't how to actually differentiate the function
Have you not covered the derivation of the C-R equations? I.e. where they come from?

The technique used in the derivation shows that f'(z_0) = u_x(x_0, y_0) + \mathbf{i} v_x (x_0, y_0) when we have f(x+iy) = u(x,y) + \mathbf{i} v(x,y) and z=x+\mathbf{i}y.
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hopelesslylost
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No, I definitely haven't covered that. Should I have already done it? My lecturer basically told me to google anything we get stuck on
(Original post by RDKGames)
Have you not covered the derivation of the C-R equations? I.e. where they come from?

The technique used in the derivation shows that f'(z_0) = u_x(x_0, y_0) + \mathbf{i} v_x (x_0, y_0) when we have f(x+iy) = u(x,y) + \mathbf{i} v(x,y) and z=x+\mathbf{i}y.
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RDKGames
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(Original post by hopelesslylost)
No, I definitely haven't covered that. Should I have already done it? My lecturer basically told me to google anything we get stuck on
Well that's not so good.

Here's the proof out of my lecture notes when I did this module.








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