If you were asked this question, how would you respond?

What is the relationship between the Cauchy-Reimann Equations and analytic functions?

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- Apr 9th 2010, 09:29 PMjzelltCauchy-Reimann Equations
If you were asked this question, how would you respond?

What is the relationship between the Cauchy-Reimann Equations and analytic functions? - Apr 9th 2010, 09:31 PMDrexel28
- Apr 9th 2010, 10:36 PMjzellt
Not sure if you're being sarcastic, but this is not a trick question.

There is going to be an essay portion on my exam, and this question is one of the possibilities...

Any thoughts? - Apr 9th 2010, 10:51 PMDrexel28
- Apr 9th 2010, 11:33 PMFailure
First note, that the man's name was R

**ie**mann,*not*Reimann.

I would say that, roughly speaking, the Cauchy-Riemann equations are the necessary and sufficient condition for a continously differentiable function $\displaystyle \mathbb{R}^2\in (x,y)\mapsto (u(x,y),v(x,y))\in \mathbb{R}^2$ to correspond via the definition $\displaystyle f: \mathbb{C}\ni x+\mathrm{i}y\mapsto u(x,y)+\mathrm{i}v(x,y)\in \mathbb{C}$ to an analytic function.

So, basically, the Cauchy-Riemann equations tell us that we can identify a subset of the continously differentiable functions $\displaystyle \mathbb{R}^2\to\mathbb{R}^2$, namely the subset of those that additionally satisfy the Cauchy-Riemann equations, with the analytic functions $\displaystyle \mathbb{C}\to\mathbb{C}$ - Apr 10th 2010, 01:08 AMPiperAlpha167
Regarding the "relationship" question, certainly there's a well-established relation of consequence between the two.

A bit devoid of details: Function f analytic in an open set S only if C-R equations hold at every point of S.

Now can we "strengthen" this relation, i.e., what can be said about the converse relation?

I think the closest we can get to the converse is by adding an additional hypothesis. - Apr 10th 2010, 01:12 AMmr fantastic
I would quote the theory that you will find in your textbook. Alternatively (if you have no textbook or class notes), I would quote from a reputable reference found using Google: eg. Cauchy?Riemann equations - Wikipedia, the free encyclopedia. Note: Holomorphic Function -- from Wolfram MathWorld

- Apr 10th 2010, 10:34 PMPiperAlpha167
I imagine it would be a good idea for me to state explicitly what I alluded to -- concerning the additional hypothesis -- in my first post.

(Perhaps it's obvious; but if not, here it is.)

The additional hypothesis is continuity of the first partial derivatives of u and v.

So then, the closest we can get to a converse of the implication given in my first post is the following:

Function f is analytic in an open set S if the first partial derivatives of f are continuous and satisfy the C-R equations at every point of S.

(Clearly, the "addition" is mediated through conjunction.)

As far as I can see, that's as close as you can get to the sought after relationship mentioned in your OP.

It's not quite a logical equivalence relation, but it's fairly close.