Complex Analysis and Transforms

  • #1
Having just gone through a section of complex analysis in a math course, I'm curious when you would actually use things like contour integration and residue theory in EE. I've been told complex analysis has all these applications in z and laplace transforms, but it seems like you only ever really need the basics like Euler's formula and algebraic manipulation of complex numbers. Contour integrals appeared in the inverse laplace and z transforms when I first learned about them, but they were brushed away as impractical when compared to using partial fractions and transform tables. Are there any cases where you would actually need some of the slightly more advanced techniques of complex analysis (e.g. residue theory), or where those techniques would provide a more practical way of doing things?

I guess I'm really wondering, when do you actually use stuff like residue theory in electrical engineering?

Answers and Replies

  • #2
I suspect you need these in advanced electromagnetics mainly for physics. You definitely don't need it in undergrad, questionable even for post grad unless you specialized in EM. I studied the EM requirement for the PHD program of U of Santa Clara which is at least a middle of the road college. They don't touch any of the complex analysis other than the basic complex number calculation and formulas.

If you want to specialize in EM, then sky is the limit, if you want to study the Classical Electrodynamics by JD Jackson..........the holy grail of EM. You definitely need complex analysis, you need intro to analysis or even a little of the real analysis. But that is not for EE. For EE, if you study Intro to Electrodynamics by Griffiths and Field and Wave Electromagnetic by Cheng, you should have a reasonable foundation already.

In real life on the job, even if you are into heavy duty analog and RF design, I doubt you ever even touch this. It's like Calculus, you never use it in real life. It just give you a lot of insight how things work and give you tools for creativity.

BTW, I take it that you finished the PDE. That is a lot more important, you get to the Fourier and Laplace transforms that is absolutely necessary for a lot of the EE subjects. Even though it is not required in undergrad, I find it give me so much more insight into EM theory. Give me a much better feel of the boundary condition.

Other math subjects like basic probability and statistics are very useful if you want to study modulation and communication theory. These are quite easy subjects. I have not study these two as I have to pick and can spend your whole life in school!!!
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  • #3
for the EE, they are there if you want to understand deeply the formulae of Fourier, Hilbert, and Laplace. residue theory can be used to understand why, for a minimum-phase filter (one with both poles and zeros in the left half of the s-plane), that the phase (in radians) is the negative of the Hilbert transform of the log-magnitude (natural log, so this would be in nepers). it's pretty hard to understand that without residue theory.

consider it a mathematical discipline. i was happy to take, as an engineering science elective, a course in Complex Variables from the math department. i came away from the course feeling better about the "magic" math we were using than when i went in.

every EE in signal processing should be solid with complex analysis (including contour integration and residue theory), probability and random numbers and random processes, matrix theory, approximation theory and numerical methods. besides the calculus, diff eq, and linear system theory.