1. ## 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?

2. 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?

3. 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?

4. 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?

5. 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 $\displaystyle \mathbb{R}^2\in (x,y)\mapsto (u(x,y),v(x,y))\in \mathbb{R}^2$ 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 $\displaystyle \mathbb{C}\to\mathbb{C}$

6. 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.

7. 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

8. 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.