[Help]Cauchy-Riemann Equations Proof

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In summary, the conversation discusses the proof of the Cauchy-Riemann equations for complex functions and how to prove the differentiability of a function that satisfies these equations. The converse of this statement requires additional assumptions and has some technical details. It is also noted that understanding the concept of linear transformations and dilation is important in understanding this proof. The book "Visual Complex Analysis" is recommended for gaining more intuition on the topic.
  • #1
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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  • #2
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.
 
  • #3
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...
 
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  • #4
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.
 
  • #5
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...
 
  • #6
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.
 

1. What are the Cauchy-Riemann equations?

The Cauchy-Riemann equations are a set of partial differential equations that specify necessary and sufficient conditions for a complex-valued function to be differentiable at a point. They are named after mathematicians Augustin-Louis Cauchy and Georg Friedrich Bernhard Riemann, who developed them in the 19th century.

2. Why are the Cauchy-Riemann equations important?

The Cauchy-Riemann equations are important because they provide a powerful tool for analyzing complex-valued functions. They are used in many areas of mathematics and physics, including complex analysis, differential geometry, and fluid dynamics.

3. How do you prove the Cauchy-Riemann equations?

The Cauchy-Riemann equations can be proved using the concept of differentiability in complex analysis. This involves showing that the limit of a difference quotient exists and is equal to a complex number independent of direction. The proof also involves using the definition of a complex derivative and the properties of limits.

4. Can the Cauchy-Riemann equations be extended to higher dimensions?

Yes, the Cauchy-Riemann equations can be extended to higher dimensions. In fact, they can be generalized to any number of dimensions by using the concept of a multidimensional complex derivative. This allows for the analysis of complex-valued functions in spaces of any dimension.

5. How are the Cauchy-Riemann equations related to the study of analytic functions?

The Cauchy-Riemann equations are closely related to the study of analytic functions, which are functions that can be represented by a convergent power series in a neighborhood of every point in their domain. In fact, the Cauchy-Riemann equations are a necessary and sufficient condition for a complex-valued function to be analytic in a given region.

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