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    Given that a function g=u+iv is differentiable on \mathbb{C}, then given that v=u^3 I need to show g(z) is constant in \mathbb{C}.

    I know that I need to use that on a given disc D(0;R) then g'(z) = 0 and g differentiable \implies g(z)=c for some constant c on D(0;R), although I'm pretty much stuck at this point...

    I initially tried using Cauchy Riemann equations out of desperation but I don't get anything useful... can anyone help me get the ball rolling?
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    What level is this?
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    (Original post by Bananas01)
    What level is this?
    Uni, although I think it'll probably be something straightforward that I just haven't spotted (as is the usual case).
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    (Original post by wanderlust.xx)
    Uni, although I think it'll probably be something straightforward that I just haven't spotted (as is the usual case).
    Just wondering because I'm doing uni maths next year.
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    It comes out quite quickly using Cauchy-Riemann equations...
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    (Original post by SimonM)
    It comes out quite quickly using Cauchy-Riemann equations...
    So g(z) = u(x,y) + iv(x,y) = u(x,y) + iu^3(x,y) and g diff \implies \dfrac{\partial u}{\partial x} = \dfrac{\partial (u^3)}{\partial y} and \dfrac{\partial u}{\partial y} = - \dfrac{\partial (u^3)}{\partial x}.

    Then what can I do? Or am I taking the wrong approach here?
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    du^3/dy = 3u^2 du/dy (all d's partial).
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    (Original post by DFranklin)
    du^3/dy = 3u^2 du/dy (all d's partial).
    :facepalm: dear god im being slow

    So basically I get (3u^4 +1) \dfrac{\partial u}{\partial x} = 0 \ \implies \dfrac{\partial u}{\partial x} = 0 \implies g'(z) = 0 \implies g(z) = c

    Buuuuut as a final question, I don't understand what happens if u^4 = -\dfrac{1}{3}? Or does this just mean g(z) = c \iff u^4 \not= -\dfrac{1}{3}?
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    Isn't in 9u^4+1? Although it doesn't really matter.

    Moving on, from context, u is real, so u^4 >=0. (If u is complex valued, f doesn't have to be constant. e.g. just take u = z).
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    (Original post by DFranklin)
    Isn't in 9u^4+1? Although it doesn't really matter.

    Moving on, from context, u is real, so u^4 >=0. (If u is complex valued, f doesn't have to be constant. e.g. just take u = z).
    yeah i missed writing out a 3 somewhere, it is in fact 9. I understand, cheers!
 
 
 
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