Complex Analysis or Complex Variables?

[PhyConnected]

Hi everyone,

I'm a Physics student going into my Junior year and I'm currently registering for my courses for the following semester and I have two options for my "complex" course, namely:

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Complex Variables

Theory of functions of one complex variable, analytic and meromorphic functions. Cauchys theorem, residue calculus, conformal mappings, introduction to analytic continuation and harmonic functions.

OR

Complex Analysis I

Complex numbers, the complex plane and Riemann sphere, Mobius transformations, elementary functions and their mapping properties, conformal mapping, holomorphic functions, Cauchys theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarzs lemma, residue theorem and residue calculus.
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The second course is a requirement for Honors Maths program, while the first one is open to all Physics students and Maths Minors.
I was wondering whether it would be worthy of taking Complex Analysis over Complex variables? (based on the course description or any similar experience)

Thanks!

 

[HallsofIvy]

That depends on you. "Complex Analysis" will be harder and include more theory and "proofs" than you really need for physics. But if you are sure you can do it and enjoy that kind of thing, go for it.

 

[PhyConnected]

That depends on you. "Complex Analysis" will be harder and include more theory and "proofs" than you really need for physics. But if you are sure you can do it and enjoy that kind of thing, go for it.

I certainly wouldn't mind doing more proofs, but does the material or extra rigor in Complex Analysis add any extra utility for a Physics major in the long-run?

For instance I took Honors multivariable Cal (Calculus on Manifolds) last year thinking that there would be limited applications, but then I eventually realized that the material played a crucial role in GR, which is a senior year course. So it seems like certain "rigorous" topics tend to become useful later.

For this case, I looked up Wiki and it says that Riemann spheres have applications in quantum mechanics.

The issue is Complex Analysis partly conflicts with a graduate optics course I'd really like to take, and in case their final exam also conflicts I'll be forced to drop the Optics course.
If the extra training isn't worthy of much, I might as well not take the risk.

Thanks again.

 

[WormBanshee]

For this situation, I'd say the difference between these two classes is different than the difference between regular and honors multivar calc. Here, there won't necessarily be more topics or applications or whatnot in the complex analysis class (actually, for computation/applications, there will likely be more useful topics in the in the complex variables class, if they do it the way I imagine they do).

The impression I have is that the Complex Variables class is more concerned with computation and calculus using complex numbers (something that as a physicist you may have to do a lot). And the Complex Analysis class will be more about developing the theory of complex numbers and their use in calculus and whatnot. A complex analysis course will mostly be concerned with proving things, while I imagine the complex variables class will be all about using complex numbers to help with computation.

Like...have you ever taken a Real Analysis course? It'll be just like that that, but with complex numbers instead of real numbers. The difference between these two classes will be similar to the difference between Calc 1&2 and Real Analysis.

 

[PhyConnected]

I took Cal1&2 instead of Hon Cal. in first year, but it still involved a good deal of Epsilon-Delta, proving properties of abstract functions and evaluating the convergence of integrals.

(my favorite one was to determine for what r this integral converges..)
\int_{0}^{\infty} \frac{x^r}{e^x} dx

I'd imagine this would be the same case for complex variables, with Complex Analysis offering even more rigor.
For instance, Stein's book was used for Complex Analysis last year and I've heard that it's a rigorous book.

I'm aware that doing more theoretical courses sacrifice a bit of the computational aspect but I believe that it'd be fairly easy to pick up those skills as you go along and there's plenty of material out there for that, while it's harder to learn pure theory all by yourself.

My primary concern is whether the extra abstraction in Complex Analysis would become significant in more advanced topics in Physics. (as Cal on Manifolds for GR and Linear Algebra for QM)

Thanks

 

[A. Bahat]

The rigorous complex analysis course may be better preparation to study Riemann surfaces, which are very important in string theory (if that's the kind of thing that floats your boat).

 

[camilus]

In my grad complex analysis class, we reviewed the entire complex variables course in a day and a half. In other words, the variables course is sort of a prereq for the analysis course. Depending on your familiarity with the complex plane, some topology, and calculus, you could probably go into the analysis class directly. It's certainly more enjoyable.

 

[mgz1985]

Complex Analysis or Complex Variables? Somethings wrong here

I work in the field of fracture mechanics. Recently, I am trying to verify the solution presented in the handbook or bible of my subject. However, I have extended my brains completely to analytically develop the equation which also makes sense physically (after all it is physics of the equation which should make sense or not). I can't understand how did the author get his solution, and so was wondering if anyone can help me?

I am putting the equation and answer below

equation : (1/i)*( (z0/(z^2-z0^2)) - (z0'/(z^2-z0'^2))

where, i = iota, z0 = x0+i*y0, z = x+i*y, z0' = conjugate(z0)

The author says equation = (y0/((z - x0)^2 + y0^2)) + (y0/((z + x0)^2 - y0^2))

Now, I have tried everything i knew, using factorization, expanding and all, but still can't develop the steps to understand this. Any help would be appreciated...

Kind regards