# Cauchy-Reimann Equations

• Apr 9th 2010, 09:29 PM
jzellt
Cauchy-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 PM
Drexel28
Quote:

Originally Posted by jzellt
If you were asked this question, how would you respond?

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

Is this a trick question?
• Apr 9th 2010, 10:36 PM
jzellt
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 PM
Drexel28
Quote:

Originally Posted by jzellt
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?

I mean, they are basically in love and married ideologically. Are you asking to show how?
• Apr 9th 2010, 11:33 PM
Failure
Quote:

Originally Posted by jzellt
If you were asked this question, how would you respond?

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

First note, that the man's name was Riemann, not Reimann.
I would say that, roughly speaking, the Cauchy-Riemann equations are the necessary and sufficient condition for a continously differentiable function $\mathbb{R}^2\in (x,y)\mapsto (u(x,y),v(x,y))\in \mathbb{R}^2$ to correspond via the definition $f: \mathbb{C}\ni x+\mathrm{i}y\mapsto u(x,y)+\mathrm{i}v(x,y)\in \mathbb{C}$ to an analytic function.

So, basically, the Cauchy-Riemann equations tell us that we can identify a subset of the continously differentiable functions $\mathbb{R}^2\to\mathbb{R}^2$, namely the subset of those that additionally satisfy the Cauchy-Riemann equations, with the analytic functions $\mathbb{C}\to\mathbb{C}$
• Apr 10th 2010, 01:08 AM
PiperAlpha167
Quote:

Originally Posted by jzellt
If you were asked this question, how would you respond?

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

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 AM
mr fantastic
Quote:

Originally Posted by jzellt
If you were asked this question, how would you respond?

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

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 PM
PiperAlpha167
Quote:

Originally Posted by PiperAlpha167
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.

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.