Scenarios where the Cauchy-Riemann equations aren't true?

1. May 6, 2013

thehangedman

Are there any scenarios where the Cauchy-Riemann equations aren't true? And if so, would there really be any difference between C^1 and R^2 in those cases?

The function: f(z, z*) = z* z doesn't solve the Cauchy-Riemann equations yet I would think is quite useful.

Couldn't we add a metric within the C^1 dimension that would give us conditions where the Cauchy-Riemann equations wouldn't work?

2. May 7, 2013

lavinia

An arbitrary continuously differentiable function from the plane into itself will no satisfy the Cauchy Riemann equations.

One requirement is that the Jacobian is complex linear: that is, it is a rotation followed by a change of scale (scalar multiplication)

The coordinate functions of an analytic function can be shown to harmonic functions.

Last edited: May 7, 2013
3. May 10, 2013

Bacle2

I'm not sure I get your notation; is your f a function of 2 complex variables? Also,
what do you mean by a difference between C^1 and R^2 ? If you refer to differentiability,
any function differentiable in one is also (Real) differentiable in the other, since C^1 and R^2
are diffeomorphic as manifolds.

For

functions of a single complex variable, f(z)=z^ , with z^ conjugation, does not satisfy C-R
anywhere. A complex analytic function must be expressed without any terms containing z^.
This means, if your function f is expressed in terms of (x,y) , then, after the coordinate change
x= (z+z^)/2 and y=(z-z^)/2i , your function is analytic if , after cleaning up , the function
can be expressed in terms of z alone.

4. May 10, 2013

thehangedman

z and z* are the same single complex variable, one the conjugate of the other. I'm wondering what would happen if we gave the plane of the single complex coordinate z a metric? Essentially, break up z into it's component parts x and y and give the x-y plane a non-Euclidean metric. If we instead had 2 complex variables for our space (C^2), then would we possibly have 3 different metrics? One for between z0 and z1, and then one for each of the x-y planes of z0 and z1 respectively? Or could we somehow wrap all three metrics into just one?