# [Help]Cauchy-Riemann Equations Proof

## Main Question or Discussion Point

Can someone give me a proof of the of Cauchy-Riemann equations? I understand how a differentiable complex function f(x,y)=u(x,y)+iv(x,y) satisfies the Cauchy-Riemann equations. How do we prove that if a complex function satisfies the Cauchy-Riemann equations, then it's differentiable. I didn't quite get the proof of Churchill's book, and I'm looking for a more elegant thought. Thanks in advance :)

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Well. I'm only familiar with the proof in the: if f is differentiable, then f satisfies cauchy riemann equations. The converse, Is actually a bit more complicated.

Eh a not so delicate way is:
$u(x + dx, y + dy) - u(x, y)= \frac{\partial{u}}{\partial{x}}dx + \frac{\partial{u}}{\partial{y}}dy$. The same can be written for v.
Then using the Equation to write $df = (\frac{\partial{u}}{\partial{x}} + i\frac{\partial{v}}{\partial{x}})(dx + idy)$. then we can see $(\frac{\partial{u}}{\partial{x}} + i\frac{\partial{v}}{\partial{x}})$ is independent of the direction you choose, hence differentiable.

Don't know if it is completely correct...

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How do we prove that if a complex function satisfies the Cauchy-Riemann equations, then it's differentiable.
This as stated is actually not true. We also need to assume that the partial derivatives are continuous.

This as stated is actually not true. We also need to assume that the partial derivatives are continuous.