Are electrodynamic problems in the complex plane relevant to real life?

Question

I have been reading Tristan Needham's excellent Visual Complex Analysis. The end of the book deals almost entirely with physics, using symmetries of conformal mappings to generalise the famous method of images technique in electrodynamics. The method of images is used in finding the electric field due to a charge when a grounded surface (such as a sphere or plane) is nearby. See http://en.wikipedia.org/wiki/Method_of_image_charges

However, the problems seem to have very little "real life" applications to me, the main problem being that the complex plane is two dimensional, whereas we live in a 3 dimensional world.

To see this problem concretely, the electrostatic force is goes like $F\sim \frac{1}{r^2}$ because the surface area of a ball of radius $r$ centred at the charge is proportional to $r^2$. However since the complex plane is 2 dimensional, a charge in the complex plane produces a field which goes like $\frac1r$. So any solution we find to a problem of this kind in the complex plane isn't relevant in 3d.

And this is my question, is there any physical application of this technique? Or is it completely irrelevant?

Answer 1

What tom's answer should convince you of is that problems with a 1d translational symmetry are essentially 2d problems, and as such you can attack them with any and all methods that are valid in 2d. That includes the use of complex analysis.

Of course, you need not use complex analysis if you don't want to. 2d vector analysis is perfectly valid instead, and I think it is under-appreciated that many complex analysis results have real vector analysis counterparts--which, for that matter, often extend to 3d and beyond.

One example of such a notion is the residue theorem. Let $q$ be the residue at a point contained in a closed curve $C$. Then if there is a meromorphic function $f$ having that residue somewhere within the region enclosed, the residue theorem tells us

$$\oint_C f(z) \, dz = 2 \pi i q$$

Consider Gauss's law. Let $E$ be the electric field and let there be a point charge with charge $Q$ inside of some surface $S$. Then Gauss's law tells us

$$\oint_S E \cdot \hat n \, dA = Q/\epsilon_0$$

These are the same basic idea. When a function has only point-source singularities in a region, the integral of the function over the bounding surface is characterized by some intrinsic number associated with each point source. (The difference here, in that the residue theorem has $2\pi$ involved while Gauss's law has no $4\pi$ could be attributed to choice of units, or to the choice of defining residues the way we do.)

Understanding the connection between complex analysis and vector analysis should make the use of the former's methods more readily understandable and less weird in contexts where such is appropriate. Personally, I would never, ever use complex analysis if I could help it. The overall mindset divorces you from vector analysis and tricks you into thinking you're doing something fundamentally different when you're not. But there is a clear mathematical relationship between the disciplines, and as such, it is not inherently weird to switch between one or the other as needed.

Answer 2

Theory of complex numbers is very useful when solving Laplace equation in 2d $$ \Delta \phi = 0 $$

If you transform harmonic function with conformal map than you again get harmonic function. Together with Riemann mapping theorem you get powerful tool for solving Laplace equation.

For example(here you are not using Riemann mapping theorem) find electrostatic field between two circles(not necessarily concentric). You can find electrostatic field between two planes and than transform this solution with Möbius trasformation to those two circles.

Or it can be used in fluid dynamics. If you try to solve potential flow than you again solve Laplace equation, so you can use these techniques to make your life easier.

As comments to the question stated there is no problem than these techniques can be used only in 2d. Often you solve problems with symmetries. For example when you want to calculate flow behind cylinder, you imagine the cylinder as infinitely long one, so basically you can forget about one dimension.

Answer 3

In three phase high voltage transformers this is a major issue. Harmonic canceling is the primary research of some power engineers. This is the same in radio transmission waves. This world has gotten extremely complex over the past 20 years as they try to understand a very hidden problem. This is starting to appear to be an unwritten law of nature that we can not put our finger on completely. So yes, these are relevant to our world, but the truth is that if we do not understand something we make our study more complex hoping to find the solution. Just remember it is good to see how others have tried to figure things out, but the solution is usually much easier than they can see.