Cauchy-Riemann Theorem Example in Physics

[entropy2008]

Homework Statement

Are there any good examples of the Cauchy-Riemann theorem (for differentiability) in physics and how it's used?

Relevant Equations

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$
$$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$

I was thinking of the wavefunction in QM but I'm not sure how it's used and when.

 

[Ishika_96_sparkles]

Please refer to the chapter on Conformal Mapping in any standard Complex analysis book (e.g. Churchill and Brown, Walter Rudin) or some engineering mathematics/Mathematical physics books (e.g. Kreyszig/ Arfken and Webber) and you would be able to find applications and problems related to two dimensional flows from the velocity potential and the stream functions defined by the vector functions u and v, respectively. Additionally, you could find examples related to the electrostatics and how it is used to solve otherwise difficult problems using regular real algebra and calculus (e.g. electric quadrupoles).

As far as QM is concerned, i am not sure if there is a sort of wave-function that conformal maps to another function in some transformed domain whose real and imaginary parts are related via the C-R equations.