Exploring the Applications of Complex Variables in Physics and Engineering

[Winzer]

I just completed a course on complex variables. I really enjoyed the application sections.
I was thinking of studying CV a little more on my own.
The question is: is it worth it to study more CV for physics and engineering?
What advantage would it give me?

Aside: I was browsing through the book "Visual Complex Analysis" Needham and found a stunning statement, at least for me: " The complex mappings that correspond to Lorentz transformations are the Mobius Transformation! Conversely, every Mobius transformation of C yields a unique Lorentz transformation of space-time." Now I may be easily entertained, but I seriously dropped my jaw at this statement. That's too amazing! What more amazing things can be done in C?

 

[Gerenuk]

Oh, haven't heard of the Lorentz transform connection.

There is something even more useful with complex numbers (apart from evaluation infinite integrals of course!). I don't recall the details, but 2D problems in fluid flow or electrostatics can be solved with conformal mapping. For that coordinates are taken as complex variables and a function of that variable can transform the problem with a complicating boundary structure into one with a very simple boundary.

For engineers and most physicists I suppose basic knowledge of contour integration and conformal mapping is sufficient.

 

[Bob S]

I have not heard of the connection between CV and the Lorentz transformation and Moibus transformations, but I confer with the previous post on the usefulness of CV in conformal mapping in solving complex problems, such as in electrostatics. Even more important is the usefulness in solving problems in EE, where impedances of circuits and systems can be either real (resistances) or reactive (capacitances and inductances). For example,
V = R + jwL -j/wC for simple series circuits. Another is the relation between watts, power factor, and volt-amps. There are similar applications in fluid flow and other ME applications. In physics, exponents can represent real (e.g., absorption and attenuation) or imaginary (e.g., energy shifts). Solution of dissipative harmonic oscillator problems require an understanding of complex variables. Causal relations, like the Kramers-Kronig relations, provide causal relations between dispersion and attenuation using complex variables. After having "grown up" using FORTRAN with COMPLEX variables as a DEFINE option, I miss it in simpler programming languages.
Bob S

 

[mikeph]

Conformal mappings in theoretical fluid flow problems is a very neat way of analysing drags and lifts on really complicated aerofoil shapes, I really enjoyed learning about them.

 

[Winzer]

Thanks for the response guys.
The things you all have mentioned I am familiar with. I guess I was thinking along the lines of more theoretical.

We covered briefly the fluid flow around objects. It's neat, but from an applications point of view isn't it too idealistic? The relations don't cover viscosity, so you get boundary layers that don't separate like they do in real life. Again, this was covered briefly so there could be more to it.