# Can complex analysis be used in classical electrodynamics?

1. Jun 17, 2011

### 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.)?

2. Jun 17, 2011

### Antiphon

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

3. Jun 17, 2011

### dingo_d

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

4. Jun 17, 2011

### Studiot

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.

5. Jun 17, 2011

### dingo_d

Great :)

Thanks for some nice insights :)