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

[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?

 

[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.

 

[Bacle2]

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?

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.

 

[thehangedman]

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.

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?