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Show that Complex-differentiable functions are always Real-diff.

  1. Feb 14, 2014 #1
    1. The problem statement, all variables and given/known data

    "Show that Complex-differentiable functions are always Real-differetiable."

    I missed the lecture where the proof of this was given and I'm trying to figure it out.

    2. Relevant equations

    Cauchy-Riemann, ux = vy, uy = -vx

    3. The attempt at a solution

    Suppose a function f(z) = u + iv is C-differentiable, so it satisfies the Cauchy-Riemann equations everywhere. So ux = vy, uy = -vx at all points.

    A function f(x,y) is R-differentiable if f(x,y) = f(x0,y0) + f(x0, y0)(x-x0)(y-y0) + O(Δx,Δy)
    and O is such that O(Δx,Δy)/(||(Δx,Δy)||) -> 0 as ||(Δx,Δy)|| -> 0.

    Firstly, my notes are a bit sketchy so it's possible my definition of R-diff is wrong.

    I'm not really sure how to use the C-R equations with this definition either. It's not as if I can just substitute some stuff in to get the required result...

    Any hints/tips/suggestions?


    Edit: Oh wait, I just realised I can apply to definition to R-diff to both u and v, since they're both functions of x and y. Will report back with where this leads.

    Edit 2: That didn't really go anywhere. I realised I should be using the C-R equations to show that f satisfies the definition of R-diff.

    I'm kind of lost.
     
    Last edited: Feb 14, 2014
  2. jcsd
  3. Feb 14, 2014 #2

    vanhees71

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    2016 Award

    Write down what it means that [itex]f[/itex] is differentiable as a complex function in the same way as you did for differentiability as a function [itex]f:\mathbb{R}^2 \rightarrow \mathbb{C}[/itex]. Then express the differentiability criterion in the sense of a complex function in terms of the real and imaginary parts.
     
  4. Feb 14, 2014 #3
    Okay, for f to be differentiable as a complex function means that the limit
    f'(z0) = lim, z->z0 [f(z) - f(z0)/(z - z0) exists.

    This has led me to believe that the question is basically a fancy way of asking for the derivation of the C-R equations, or at least the first half of it.

    Is that correct? I can express z, z0 in terms of x,y etc, and then just apply the conditions of R-differentiability to the real and imaginary parts, and I'll eventually just end up with the Cauchy-Riemann equations.
     
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