# Usefulness of complex analysis for the physical sciences?

## Main Question or Discussion Point

Hi, I am interested in taking a complex analysis course. How useful is it to the physical sciences?

I spend most of my time with electronics and lasers but i end up using complex analysis in some way all the time. Anything that can be described via a sine wave (harmonic motion) can also be expressed as a complex function, this happens quite often for me.

Cauchy-Riemann conditions are somewhere else you'll run into complex numbers. I'm sure there are a million more examples.

Now while i do use complex analysis quite often i never took a coarse in it and i am surviving quite fine. However, i would say that if it looks interesting go for it! It will definitely be useful at some point.

I spend most of my time with electronics and lasers but i end up using complex analysis in some way all the time. Anything that can be described via a sine wave (harmonic motion) can also be expressed as a complex function, this happens quite often for me.

Cauchy-Riemann conditions are somewhere else you'll run into complex numbers. I'm sure there are a million more examples.

Now while i do use complex analysis quite often i never took a coarse in it and i am surviving quite fine. However, i would say that if it looks interesting go for it! It will definitely be useful at some point.
you're probably just using the complex expression of the fourier series which isn't really complex analysis.

anyway having just finished my complex "variables" final, an applied complex analysis class, i can tell you that the only useful thing i got out of it was the calculus of residues and laurent series. so yea. skip that class and just learn how to use/do those two things. of course for both those things you'll need to understand things like analytic functions and the cauch riemann equations. we spent half a semester covering stupid things like how to take the mod of a complex number so just skip all that.

diazona
Homework Helper
I'm finishing up my first year of graduate school in physics. I took one course in complex analysis as an undergrad, and to be honest, I don't think anything I learned in that class has ever been useful to me. People keep saying complex analysis is incredibly useful in physics but I'm still waiting to see evidence of that. Sure, I use complex numbers all the time, but it's generally just arithmetic, complex conjugation, complex exponentials (like exp(i*x) = cos(x) + i*sin(x)), and Taylor series (occasionally Laurent series) of functions, usually real ones. None of which I needed a complex analysis course to understand.

In retrospect, I feel like my time would have been much better spent studying group theory...

I'm finishing up my first year of graduate school in physics. I took one course in complex analysis as an undergrad, and to be honest, I don't think anything I learned in that class has ever been useful to me. People keep saying complex analysis is incredibly useful in physics but I'm still waiting to see evidence of that. Sure, I use complex numbers all the time, but it's generally just arithmetic, complex conjugation, complex exponentials (like exp(i*x) = cos(x) + i*sin(x)), and Taylor series (occasionally Laurent series) of functions, usually real ones. None of which I needed a complex analysis course to understand.

In retrospect, I feel like my time would have been much better spent studying group theory...
did you do the residue theorem in that class?

diazona
Homework Helper
Yes I did... I'm looking back at my textbook's table of contents and it includes basic complex arithmetic (which was a review), analytic functions, contour integrals, power series, residues and their applications to integration, and mappings of the complex plane. I think we covered more or less all of that in the class I took.

Well for me i didn't take a course in complex analysis but i do have to use parts of it every so often (I am a plasma physicist)

I think it depends on where you plan to go with physics and what area you wish to get into. There are some areas where it could be something that is very helpful and others where it is not needed.

I would say look into what you plan to be and see if the course would be relevant.

hope this helps.
~N~

you're probably just using the complex expression of the fourier series which isn't really complex analysis.

anyway having just finished my complex "variables" final, an applied complex analysis class, i can tell you that the only useful thing i got out of it was the calculus of residues and laurent series. so yea. skip that class and just learn how to use/do those two things. of course for both those things you'll need to understand things like analytic functions and the cauch riemann equations. we spent half a semester covering stupid things like how to take the mod of a complex number so just skip all that.
Good to know :) I actually saw residue theorem, Laurent series, and Cauchy-Riemann in a class that was called mathematical methods for physicists so if something like that is offered it might be more worthwhile. The class I took picked out what was deemed to be useful out of complex analysis while also getting other things such as tensor analysis. /shrug

Good to know :) I actually saw residue theorem, Laurent series, and Cauchy-Riemann in a class that was called mathematical methods for physicists so if something like that is offered it might be more worthwhile. The class I took picked out what was deemed to be useful out of complex analysis while also getting other things such as tensor analysis. /shrug
yea just look up indented paths and how to do trig integrals using the residue theorem and you're golden. there's some stuff in the back of my complex analysis book about conformal mappings but i have yet to check that out so i don't know if it's useful or not.

I don't know how useful it is in "doing" physics, but I think learning complex analysis deeply fills a hole that every human being has in their heart after taking just real calculus, whether they know it or not. It's like standing in front of the Louvre and deciding whether to go in: it's not a decision that should be taken on the basis of material usefulness.

I don't know how useful it is in "doing" physics, but I think learning complex analysis deeply fills a hole that every human being has in their heart after taking just real calculus, whether they know it or not. It's like standing in front of the Louvre and deciding whether to go in: it's not a decision that should be taken on the basis of material usefulness.
That's a fabulous analogy, and I now will be planning on adding complex analyses to my upcoming curriculum. Eventually. :)

I don't know how useful it is in "doing" physics, but I think learning complex analysis deeply fills a hole that every human being has in their heart after taking just real calculus, whether they know it or not. It's like standing in front of the Louvre and deciding whether to go in: it's not a decision that should be taken on the basis of material usefulness.
what in the hell makes you say so?

On the one hand, I've heard my teacher praising Complex Analysis for its beauty and its simplicity compared to real analysis (of course difficulty is somewhat subjective).

On the other hand, I have heard that Complex Analysis books have a lot of computation, even in the proofs. Spivak's Calculus introduces the very basics of complex analysis fairly nicely, yet his proof of the fundamental theorem of algebra looks horrible.

Consider the two following important results in complex analysis (stated imprecisely of course):

1) The integral of a function over a closed curve is determined only by its behavior at the interior singularities. (Cauchy's Integral Formula)
2) A function takes on every value (except possibly one) infinitely often in any neighborhood of an essential singularity. (Piccard's Theorem)

If that's not beauty, I don't know what is.

And if you go through a full development of complex analysis then the Fundamental Theorem pops out as an immediate consequence of the brilliantly simple Louiville's Theorem: every non-constant function that is entire (differentiable everywhere on the complex plane) must be unbounded, a striking departure from the real case.

Molu

Try Needham's Visual Complex Analysis. It may not be entirely rigorous, but it's very exciting.

Molu

For me, its just method of residues which has been useful so far. It comes up when taking the actual inverse Laplace transform, not using a table. I need to do this when solving certain pde's in fluid mechanics. I'm sure there are many other uses for complex analysis. You could either take it now for credit and risk forgetting it, or audit/teach yourself it later when you need it.