# Can complex analysis be used in classical electrodynamics?

• dingo_d
In summary, the conversation discusses the use of theorems in complex analysis in electrodynamics, specifically in the context of the Cauchy-Goursat theorem and its relation to the irrotational electric field and Laplace's equation. The use of conformal mapping in solving these equations is also mentioned.
dingo_d
The title may be a bit vague, so I'll state what I am curious about.

Since complex field is 'extension' to the real field, and in electrodynamics we use things like Stokes theorem, or Gauss theorem, that are being done on real field (differential manifolds and things like that, right?), can we use theorems in complex analysis, and say that some feature can be described because of that?

An example.

In complex analysis there is famous Cauchy-Goursat theorem which states that if we have some analytical function on a convex set, and if we have some closed path in that set the integral:

$$\oint_\gamma f(z)dz=0$$

And in electrostatics we have the irrotational electric field

$$\vec{\nabla}\times \vec{E}=0$$ which if we use Kelvin-Stokes becomes:

$$\oint \vec{E}\cdot d\vec{\ell}=0$$.

Is this just a generalization of CG theorem and could we use all the theorems in complex analysis, or should we need to be more careful (like with analytic functions etc.)?

It's done a lot and comes up naturally in the frequency domain where all fields are complex already.

So it's legitimate to say: this follows from complex analysis, like in above example? I'm not making any broad assumptions?

Laplace's Equation, and its solution, plays quite a significant role in Electrodynamics.

One method of solution is to use conformal mapping which yields 2D maps of the fields, in a similar manner to fluid flow nets or stress fields.

Great :)

Thanks for some nice insights :)

## 1. Can complex analysis be used in classical electrodynamics?

Yes, complex analysis can be used in classical electrodynamics. In fact, it is an essential tool for solving many problems in this field. Complex analysis allows us to represent electromagnetic fields in terms of complex functions, making calculations and solutions more efficient and elegant.

## 2. What is the relationship between complex analysis and classical electrodynamics?

Complex analysis is a branch of mathematics that deals with functions of complex numbers. It is closely related to classical electrodynamics, which is the study of the interactions between electric and magnetic fields. Complex analysis provides powerful techniques for solving problems in classical electrodynamics.

## 3. How does complex analysis help in understanding electromagnetic phenomena?

Complex analysis helps in understanding electromagnetic phenomena by providing a mathematical framework for analyzing and solving problems involving electric and magnetic fields. It allows us to represent electromagnetic fields as complex functions, which can then be manipulated and analyzed using various techniques.

## 4. What are some specific applications of complex analysis in classical electrodynamics?

Some specific applications of complex analysis in classical electrodynamics include calculating the electric and magnetic fields of charged particles, analyzing the behavior of electromagnetic waves, and solving boundary value problems for electric and magnetic fields. Complex analysis is also used in the study of electromagnetic radiation.

## 5. Are there any limitations to using complex analysis in classical electrodynamics?

While complex analysis is a powerful tool in classical electrodynamics, it does have some limitations. It may not be applicable to highly nonlinear systems or problems involving complex geometries. In addition, complex analysis is only one aspect of understanding electromagnetic phenomena and should be used in conjunction with other mathematical and physical methods for a complete understanding of the subject.

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