# Cauchy-Riemann Theorem Example in Physics

• entropy2008
In summary, the conversation discusses the use of wavefunctions in quantum mechanics and its applications in solving problems related to two-dimensional flows and electrostatics. It is suggested to refer to specific chapters in complex analysis and engineering mathematics books for further understanding and examples. There is also a mention of the possibility of a conformal map between wavefunctions in QM and functions in a transformed domain.

#### 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.

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.

TSny and entropy2008

## 1. What is the Cauchy-Riemann theorem?

The Cauchy-Riemann theorem is a fundamental theorem in complex analysis that provides necessary and sufficient conditions for a complex-valued function to be differentiable at a point. It states that a function is differentiable at a point if and only if it satisfies a set of partial differential equations known as the Cauchy-Riemann equations.

## 2. How is the Cauchy-Riemann theorem applied in physics?

The Cauchy-Riemann theorem has numerous applications in physics, particularly in the study of fluid dynamics and electromagnetism. In fluid dynamics, the theorem is used to analyze the behavior of complex velocity fields, while in electromagnetism, it is used to study the behavior of complex electric and magnetic fields.

## 3. Can you provide an example of the Cauchy-Riemann theorem in physics?

One example of the Cauchy-Riemann theorem in physics is its application in analyzing the flow of an ideal fluid. The velocity field of an ideal fluid is described by a complex function, and the Cauchy-Riemann equations can be used to determine the conditions for the fluid to be incompressible and irrotational.

## 4. How does the Cauchy-Riemann theorem relate to the conservation of energy in physics?

The Cauchy-Riemann theorem is closely related to the conservation of energy in physics. This is because the Cauchy-Riemann equations imply that the real and imaginary parts of a complex function, which correspond to the potential and stream functions in fluid dynamics, respectively, must satisfy the Laplace equation. This equation is a fundamental equation in physics that describes the conservation of energy.

## 5. Are there any limitations to the Cauchy-Riemann theorem in physics?

While the Cauchy-Riemann theorem is a powerful tool in analyzing complex functions in physics, it does have some limitations. One limitation is that it only applies to functions that are differentiable at a point, which may not always be the case in complex systems. Additionally, the theorem is limited to two-dimensional systems and does not apply to higher-dimensional systems.