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?

 

[blue_raver22]

I can confirm that there are indeed scenarios where the Cauchy-Riemann equations may not hold true. These equations are based on the assumption that a complex function is differentiable at a point, and this is not always the case.

One example is when the function is not analytic, meaning it is not infinitely differentiable at all points in its domain. In this case, the Cauchy-Riemann equations may not be satisfied, and the function cannot be described using the traditional methods of complex analysis.

Another scenario where the Cauchy-Riemann equations may not hold true is when dealing with functions that are not holomorphic. A holomorphic function is one that is differentiable at all points within its domain, and thus satisfies the Cauchy-Riemann equations. However, there are many functions that are not holomorphic, such as those with singularities or branch points, and the Cauchy-Riemann equations would not apply in these cases.

Regarding the question of whether there would be a difference between C^1 and R^2 in these cases, it is important to note that the Cauchy-Riemann equations are specific to the complex plane and cannot be extended to other dimensions. Therefore, the concept of C^1 and R^2 would not be applicable in these scenarios.

In regards to the function f(z, z*) = z* z, it is true that this function does not satisfy the Cauchy-Riemann equations. However, it is still a useful function in certain contexts, such as in quantum mechanics and statistical mechanics. In these cases, the function may be used in conjunction with other mathematical tools to describe physical phenomena.

In conclusion, while the Cauchy-Riemann equations are a fundamental tool in complex analysis, there are indeed scenarios where they may not hold true. As scientists, it is important to understand these limitations and use appropriate mathematical tools to accurately describe and analyze complex systems.