[Help]Cauchy-Riemann Equations Proof

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Discussion Overview

The discussion revolves around the proof of the Cauchy-Riemann equations, specifically focusing on the converse: proving that if a complex function satisfies the Cauchy-Riemann equations, then it is differentiable. Participants express varying degrees of familiarity with existing proofs and seek a more elegant or straightforward approach.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests a proof for the converse of the Cauchy-Riemann equations, indicating a desire for clarity beyond existing texts.
  • Another participant notes that they are only familiar with the proof that if a function is differentiable, it satisfies the Cauchy-Riemann equations, suggesting that the converse is more complex.
  • A participant presents a method involving partial derivatives and the differential of the function, expressing uncertainty about its correctness.
  • One participant asserts that the statement about differentiability given the Cauchy-Riemann equations is not true without the assumption of continuous partial derivatives.
  • After acknowledging the need for continuous partial derivatives, another participant contemplates using the definition of the derivative for a straightforward proof but doubts its effectiveness.
  • A later reply discusses the relationship between linear transformations and complex differentiability, suggesting that understanding this connection is crucial for grasping the proof's requirements.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the proof's approach or the necessity of continuous partial derivatives, indicating that multiple competing views remain regarding the proof of differentiability from the Cauchy-Riemann equations.

Contextual Notes

Participants highlight the importance of continuity of partial derivatives in the context of the proof, which remains an unresolved aspect of the discussion.

kostas230
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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:
[itex]u(x + dx, y + dy) - u(x, y)= \frac{\partial{u}}{\partial{x}}dx + \frac{\partial{u}}{\partial{y}}dy[/itex]. The same can be written for v.
Then using the Equation to write [itex]df = (\frac{\partial{u}}{\partial{x}} + i\frac{\partial{v}}{\partial{x}})(dx + idy)[/itex]. then we can see [itex](\frac{\partial{u}}{\partial{x}} + i\frac{\partial{v}}{\partial{x}})[/itex] is independent of the direction you choose, hence differentiable.

Don't know if it is completely correct...
 
Last edited:
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.
 
A. Bahat said:
This as stated is actually not true. We also need to assume that the partial derivatives are continuous.

My bad. Sorry :shy:
OK, let's assume that its partial derivatives are continuous. How do we prove that it's differentiable? I was thinking a straightforward proof using the definition of the derivative, but I don't think it's going to give me any good result...
 
Well, the big idea here is that a linear transformation acts on R^2, just like multiplication by a complex number does if it satisfies the Cauchy-Riemann equations. So, basically, it's condition for a linear transformation to be a dilation (i.e. a rotation composed with rescaling). That's probably the main thing to understand. You're just applying this fact to the differential of a function from R^2 to R^2.

Going from complex differentiability to this condition isn't too bad. The converse has some technical details to prove rigorously. If you want more intuition, read Visual Complex Analysis.
 

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