# Is complex analysis necessary for electrical engineering?

1. Aug 30, 2014

### Alexander2357

I am and EE and CS double major and I am not sure whether to take complex analysis or not.

Linear algebra, multivariable calculus, differential equations and probability are compulsory but complex analysis and stochastic processes are optional, so I am wondering whether I should take them or not.

In what areas of EE is complex analysis used?

2. Aug 30, 2014

### Simon Bridge

You'll first meet complex analysis in linear networks where you handle complex impedences. It's used anywhere you have a signal (represented by a phasor in the complex plain) to contend with.

When I did it, you only needed to know what a complex number was and a bit of algebra covered in the coursework. That'll be why it's optional - most people can figure out the stuff they need for the course.

I'd say: only take it if you enjoy it. If you don't enjoy it, you are unlikely to end up working in a direction where it's needed in so much depth anyway. But you may want to be Mr Ubergineer Polymathguy.

3. Aug 30, 2014

### Alexander2357

I studied complex numbers in high school and one of the topics in multivariable calculus at my university covers complex numbers a little bit further.

I have been interested in complex analysis since high school but to take that course I would have to take proof-based real analysis, which I don't need in either EE or CS (also, I would prefer to avoid that class...). I will think about it further. Real analysis will be tough as I am not good at writing proofs but all the upper maths classes have real analysis as a prerequisite (except some statistics and operations research classes).

If I don't do complex analysis, then that means I don't need to do real analysis, so this frees up two slots in my timetable.

Thank you very much for your answer. I always wondered how complex analysis is used in EE and your answer explained it very well.

Last edited: Aug 30, 2014
4. Aug 30, 2014

### Staff: Mentor

If you study AC circuits, and especially three phase circuits, then complex numbers are imperative.

But I tend to be suspicious of academic courses, and what they may mean by "complex analysis." The things that are imperative you can learn in a few hours. I recommend that you scrutinize the syllabus.

5. Aug 30, 2014

### thegreenlaser

It's probably not worth doing the real analysis to complex analysis sequence if you're not interested in definition-theorem-proof style math. My experience has been the same as Simon Bridge: it's really not necessary if you don't want to take it.

If you're interested in it, basic complex analysis is something you probably won't regret taking. The big thing is to learn about integrals in the complex plane and all the key theorems surrounding them. When you've learned that stuff, things like the inverse Laplace transform and some of the methods used in control systems will make a lot more sense (I think z-transforms also made more sense? I can't remember for sure). Complex analysis is also handy because it gives you some tricks to calculate real-valued integrals.

The thing is, electrical engineers use complex numbers and complex functions all the time. So it certainly doesn't hurt to dig a little deeper into the theory of complex numbers and complex functions. Some more advanced materials will probably even assume that you know this kind of stuff.

If you don't want to do two rigorous courses in analysis (which is understandable), try to look for some sort of "mathematical methods" course, like what most physics majors would take. A course like that would usually cover stuff like partial differential equations (also useful) along with the really interesting and useful bits from complex analysis.

6. Aug 30, 2014

### pasmith

Knowing some complex analysis will be of assistance when you come to study Laplace and Fourier transforms; aside from anything else, you may actually be able to use the inverse laplace transform formula rather than relying on tables.

7. Aug 30, 2014

### Alexander2357

Here is the syllabus:

• The topology of the complex plane
• convergence of complex sequences and series
• analytic functions, the Cauchy-Riemann equations, harmonic functions and applications
• contour integrals and the Cauchy Integral Theorem
• singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping
• aspects of the gamma function.

Last edited: Aug 30, 2014
8. Aug 30, 2014

### thegreenlaser

The complex number theory used in AC circuits is pretty basic. You're just doing algebra with complex variables, which is something any electrical engineer should be familiar with. I didn't find complex analysis helped much there because it's already so simple.

The complex analysis I took (and I assume it's the same for most people) was a lot more focused on the calculus of complex functions: taking the derivative with respect to a complex variable, doing path integrals in the complex plane, and all the surprising and useful theorems that come out of that. That stuff won't help you with AC circuits, but it will help whenever calculus of complex functions is used.

9. Aug 30, 2014

### thegreenlaser

That definitely covers the key concepts (and more). Again, I'd suggest checking if your school offers a "mathematical methods" course, either for engineering or physics students. You really don't need the depth of a full definition-theorem-proof style course.

10. Aug 30, 2014

### homeomorphic

A lot of EE students would probably benefit quite a bit from chapter 1 of a complex analysis book. I remember I had enough knowledge of complex numbers to get by, but when I read Chapter 1 of Visual Complex Analysis, it made so much more sense, and I was able to understand the basics so much more vividly. I got hooked and kept wanting to read more about the deeper and deeper justifications of Euler's formula in the book, which is essentially the reason I started reading it.

Of course, then, I took it too far and ditched EE to became a failed mathematician.

11. Aug 30, 2014

### Alexander2357

Thank you very much for your very helpful answers, thegreenlaser.

Unfortunately, that kind of subject isn't available at my university. Physics majors have to take almost the same subjects as applied mathematics majors.

There is a subject called engineering mathematics but it is basically vector calculus and differential equations combined, with no complex analysis.

12. Aug 30, 2014

### Alexander2357

I think I will read that book too, I already have it.
I was initially planning on double majoring in math and CS, then changed my mind to EE and CS with 7 maths electives and it looks like I am going to go down to 6 maths electives...

13. Aug 30, 2014

### AlephZero

That looks like it is leaning towards being a "pure math" course, especially it is it proof-based and has "real analysis" as a prerequisite.

For EE, control system theory, experimental vibration measurement in ME, etc, if would probably be better to focus on the applications, e.g. singularities, residues, contour integrals, etc, and how they relate to the physics and engineering, rather than on the mathematical proofs.

But obviously you can only take the courses that are offered at your college.

14. Aug 30, 2014

### chill_factor

try to take a math class in physics or electrical engineering, rather than an actual math class. you need arithmetic - using known mathematical principles to calculate quantities. you don't need new math.

15. Aug 31, 2014

### thegreenlaser

That's unfortunate... Personally, I would just study some less rigorous complex analysis on your own then. It's probably not worth doing the "mathematician" approach unless you're really interested in it.

16. Aug 31, 2014

### fisicist

The thing is that there is nothing else than the "mathematician approach". All you really need to know is the Cauchy Riemann PDE together with the Liouville theorem, the residue theorem and perhaps the homotopy version of Cauchy's theorem. If you assume the complex derivative of a holomorphic function to be continuous (which is of course true, but takes some time to be proven), you can do all this in three hours, we did this in (real) Analysis 2. Nevertheless, I did not regret attending the complex analysis lecture. It's such a wonderful subject, it is impossible not to fall in love with it!

You find a shortcut in The Road To Reality by Roger Penrose.

17. Aug 31, 2014

### FactChecker

As others have said, try to get into an applied complex analysis course for engineers rather than a pure math course. The benefits will be in the areas of electrostatic potential theory, control systems and feedback. I don't think it is possible to really understand those subjects without it.

18. Aug 31, 2014

### thegreenlaser

What I meant by "mathematician approach" is the amount of detail and rigor that you would see in a class aimed at math students. For most engineering students, it's not worth spending a lot of time studying the rigorous proof of each theorem. They just need to know what the key theorems are and how to apply them.

19. Aug 31, 2014

### mathwonk

I am a mathematician, so take my comments in that spirit. The relevance of complex variables to electricity seems to be the fact that if w = f(z) is a differentiable function of a complex variable z, then both real and imaginary parts U,V of the function f(z) = U(z) + iV(z), will satisfy Laplace's equation ∂^2 U/∂x^2 + ∂^2 U/∂y^2 = 0, where z = x+iy. So the usefulness of this mathematics is because of the appearance of that equation in the study of electrical fields. See Feynman's lectures, volume II, chapter 7-2, for a short discussion. I apologize if I have misunderstood.

I agree that the facts are more useful than the proofs, but I myself have trouble remembering facts for which I do not know at least roughly the proofs. I.e. when I want to remember the facts, I rethink the proofs. In my day, the classic text on applied complex analysis was churchill, later rewritten with/by Brown: it may still be too theoretical for engineers, so take a look yourself, and you be the judge:

https://www.amazon.com/Complex-Vari...d=1409525587&sr=1-1&keywords=ruel+v+churchill

Last edited by a moderator: May 6, 2017
20. Sep 1, 2014

### jasonRF

As others noted, complex analysis can be useful. For me it has mostly been useful for dealing with nasty integrals, especially those that arise in Fourier analysis and probability calculations. Complex analysis was mandatory for my graduate work in EE, but many students learned it the first semester of grad school via an applied math course taught by the engineering school.

In my experience stochastic processes (at least the way a typical EE department teaches it) is MUCH more useful. After all, actual physical systems all have noise, and such a course allows you to develop the tools you need to think about noisy signals.

jason