Proving Holomorphic Functions are Analytic: The Role of Cauchy Riemann Equations

[Mappe]

I was told an analytic complex functions has the same derivation value at z0 (random point) however you approach z0. The cauchy riemann eq. shows that z0 has the same derivate value from 2 directions, perpendicular to each other. However, at least some real functions can have the same derivate value in (x0,y0) (random point) approached from 2 directions, without having the same value in another, third direction. So, how does the cauchy riemann eq. prove that its the same from every angle, and not only equal from x and y directions? Hope you understand my question, thnx.

 

[fzero]

There's an additional condition that the real and imaginary parts be continuously differentiable functions of x and y. This is usually stated in any reference on the CR equations, so you might want to recheck your source. If the derivatives in the CR equations don't exist, it's pointless to try to apply them.

 

[Mappe]

Yes, I know about that condition, but how does that imply that the derivate value in one point is independent of the direction in which it is approached? What does being continuously differentiable have to do with that?

 

[fzero]

Yes, I know about that condition, but how does that imply that the derivate value in one point is independent of the direction in which it is approached? What does being continuously differentiable have to do with that?

If a derivative depends on the path, that derivative does not exist, and the function is not differentiable. If a function is differentiable, all derivatives exist.

 

[kloptok]

So the statement is basically that given an analytic function f=u+iv, where u and v have existing derivatives* that are continous, u and v will satisfy C-R's equations?*Since the derivatives exist, they must necessarily be the same regardless of which direction you approach the point from.

 

[fzero]

So the statement is basically that given an analytic function f=u+iv, where u and v have existing derivatives* that are continous, u and v will satisfy C-R's equations?


*Since the derivatives exist, they must necessarily be the same regardless of which direction you approach the point from.

If f is analytic, u and v are continuous and differentiable. The use of "where" is a bit misleading, since it follows from analyticity and is not an additional assumption.

 

[BackEMF]

What you're asking comes down to seperating the necessary and sufficient conditions. As you have said in real functions of a real variable, a third direction may yield a different limit and hence the function in question may not be differenentiable. But you agree that if the function is to be differentiable, it MUST satisfy the C-R equations? In other words, they are necessary for a function to be analytic. They are not sufficient.

The situation is:

1. Necessary condition: f(z) is analytic => C-R equations satisfied (True)
2. Sufficient condition: C-R equations statisfied => f(z) is analytic (False)

The first one is the necessary condition and it must hold if f(z) is analytic.

The second one is false in general. However if we also assume that all four partials derivatives of u(x,y) and v(x,y) exist and are themselves continuous, then the second condition is true. The proof of this is a good bit more tricky than the necessary proof that you have referred to. It involves the mean value theorem, and this applies to continuous, differentiable functions.

Hence, if these extra conditions are valid we have a two-way implication (aka a necessary and sufficient condition):

f(z) is analytic <=> C-R equations satisfied

 

[mathwonk]

yes you have it backwards. the fact that the limits are the same from all directions is an assumption about holomorphic functions. then the cauchy riemann equations are a consequence of that assumption. i.e. it follows that the limits are the same from the two x and y directions. that tautological fact is the cauchy riemann equations.the converse of this is less trivial. i.e. if we assume instead the cauchy riemann equations hold on an open set, and also that the partials are continuous, then it follows that the function is holomorphic. i.e. in the presence of continuity of the partials, the equality of the limits in two directions implies the equality in all directions. this takes a little more work to prove.