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Maybe_Memorie

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## Homework Statement

"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.

## Homework Equations

Cauchy-Riemann, u

_{x}= v

_{y}, u

_{y}= -v

_{x}

## The Attempt at a Solution

Suppose a function f(z) = u + iv is C-differentiable, so it satisfies the Cauchy-Riemann equations everywhere. So u

_{x}= v

_{y}, u

_{y}= -v

_{x}at all points.

A function f(x,y) is R-differentiable if f(x,y) = f(x

_{0},y

_{0}) + f(x

_{0}, y

_{0})(x-x

_{0})(y-y

_{0}) + 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 into get the required result...

Any hints/tips/suggestions?

Edit: Oh wait, I just realized 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 realized I should be using the C-R equations to show that f satisfies the definition of R-diff.

I'm kind of lost.

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