Complex Analysis Courses or Complex Variable Courses?

[Jonathanos]

Hello,

My university offers a couple Complex Analysis courses, among them there is one with the following description:

Introduction to complex variables:
"substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous."
(Content) "Differentiation and integration of complex-valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, applications in ideal fluid dynamics. This corresponds to Chapters 1-9 of Churchill & Brown. "

Complex I (Complex):
"This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs."
(Content)"Review of analysis in R2 including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, linear fractional transformations; Cauchy’s theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem. "

For a little background, these two math classes have the same pre-reqs. They both require Real Analysis, which I have taken. I am majoring in physics and math but I want to go to grad school for physics. I understand that math classes can be different than what a physicist needs. I have been told that complex analysis is a great class to take if possible for physics, however, since I have the choice between these two, I would just like some guidance of which one might be better for me and why.

TLDR: Complex Analysis more applications or more proofs for physics major.

Thank you!

 

[member 587159]

This seems like a no-brainer if you want physics/applications. Pick the first one.

 

[Vanadium 50]

substantial attention to applications in science

Why would you not pick this one?

 

[Jonathanos]

Why would you not pick this one?

Because I am interested in theory and thought it might be useful to know this kind of math if I want to be a theoretical physicist. However, I'm not completely sure if I am correct. If it does not matter which one I take, I would rather take the first option.

 

[Dr Transport]

Unless you do a PhD in Mathematical Physics, a theorem-proof class in my opinion would not be that helpful. I'm a theoretician and have not had a need for that level of mathematical rigor.

 

[mathwonk]

I am not a physicist, I am a mathematician, but I have taught both those courses. I would imagine that for you the applied course is more useful. I.e. you would probably rather understand how to use complex analysis than how to prove the theorems.

 

[FactChecker]

My Ph.D. thesis research is in the field of complex analysis. For a physics major, you do not want to spend all your time learning the minutia of formal proofs. That time is better spent in the first course, learning how it is applied.

On the other hand, if you take the second class, you might see some things that make the subject so amazing -- like a gift from God. Then you might turn into a theoretical mathematician. That is a horrible fate. ;>)

 

[Jonathanos]

My Ph.D. thesis research is in the field of complex analysis. For a physics major, you do not want to spend all your time learning the minutia of formal proofs. That time is better spent in the first course, learning how it is applied.

On the other hand, if you take the second class, you might see some things that make the subject so amazing -- like a gift from God. Then you might turn into a theoretical mathematician. That is a horrible fate. ;>)

No thank you, my professor for a math class is a Complex Analyst the first day of class he mentioned "So you are all here because you like math?... Why do you hate yourselves?" Why would I ever want to associate myself with such likable people? That indeed would be a horrible fate :P

 

[FactChecker]

No thank you, my professor for a math class is a Complex Analyst the first day of class he mentioned "So you are all here because you like math?... Why do you hate yourselves?" Why would I ever want to associate myself with such likable people? That indeed would be a horrible fate :P

He was probably doing you a favor.

 

[mpresic3]

I doubt that the theory course would look any better on a transcript that the applied course for graduate study in physics, but I have never been on an admissions committee so you might want to check this with physics theoreticians at your school.
Alternatively, you might think you want to get the theory so you are better prepared for graduate work in physics. I would suggest the applied one would be more valuable.
The only case that could be made is if you wanted to do graduate study in a branch of physics where the theory of complex variables is useful. These areas do exist, but you should ask yourself, do you really know right now what math will be required for your graduate research.
Also should you need the theory, Is this the last chance you have to get the theory coursework. Won't the graduate school you attend have a similar course, (should you need it)? Your future thesis advisor or graduate advisor would amost certainly OK it.

 

[Jonathanos]

I doubt that the theory course would look any better on a transcript that the applied course for graduate study in physics, but I have never been on an admissions committee so you might want to check this with physics theoreticians at your school.
Alternatively, you might think you want to get the theory so you are better prepared for graduate work in physics. I would suggest the applied one would be more valuable.
The only case that could be made is if you wanted to do graduate study in a branch of physics where the theory of complex variables is useful. These areas do exist, but you should ask yourself, do you really know right now what math will be required for your graduate research.
Also should you need the theory, Is this the last chance you have to get the theory coursework. Won't the graduate school you attend have a similar course, (should you need it)? Your future thesis advisor or graduate advisor would amost certainly OK it.

I think I will just take the application based one. I think it is nice to first learn how to use it and then learn why it actually works: taking Calculus before Real Analysis makes more sense to me now that I think about it a little more. The only reason I was planning on taking the theoretical class was so that if I wanted to take other advanced courses (for example Partial Differential Equations or Functional Analysis) that required the theoretical class I would be able to take them sooner. However, realistically, I doubt I will have the time to get that far in math before graduate school. I also need to focus on my physics classes, so as much as I might dislike the idea, I need to eventually stop taking math courses :/

 

[dRic2]

In my master in nuclear engineering a math course in complex analysis was mandatory. Although it was designed for engineers it was very theoretical with lots of rigorous mathematical proofs and very few practical applications. The description of "Complex I" is very similar to the one I took. It was hard, but I must confess it is probably the best course that I ever had. It helped me tremendously with understanding other materials and gave me the "method" to study new math topics on my own. It might be an Overkill but a more rigourous course can give you more prospective. For example later in my studies I took a course on plasma physics for fusion applications. When I was studying plasma oscillations I ran into a well known problem (solved by Landau) which basically arose from a mis-use of the Fourier Transform. I could have understood it even without all the rigourous stuff that I when through in my complex analysis course, but I have to say that it gave me a new perspective to see things.

As I said it might not be necessary, but if you have enough time I would try.

 

[hutchphd]

Is the same prof teaching the two courses? Sometimes the "applied" profs are not as good...and a good professor can change a life ( or so I used to tell myself when I didn't feel the burn ). Might be worth all the proofs.

 

[mpresic3]

I did not realize you were taking the class as a prequisite for Functional Analysis or Partial Differential Equations. I grant these courses are important to theoretical physics in grad school.
As hutchphd writes, a good prof can change your life. But, there are good "applied" profs too. You are right to think there is only a finite amount of math courses at the upper undergrad level you can take, as these are demanding and you will probably need/want to include undergraduate research, studying for GRE's, choosing Letters of Recommendations etc that grad schools will need.
I do not think you should stop taking math courses, but you may consider whether to defer them until grad school if they are necessary for your work.

In my experience, (I did take functional analysis as a senior undergraduate, out of Reid/Simon, and Lebesgue integration out of Bartle, although Reid and Simon was barely started, since it was a multivolume work), the functional analysis helped me communicate with mathematician friends of mine, years later after grad school.
I feel the intro complex variables would be more valuable to me when I needed a mathematical physics course as part of a core requirement in first year grad school. Others on the forum may disagree.

You probably can get better advice from your school undergraduate advisor in the physics or math departments as to your directions to promote your educational goals.

 

[Jonathanos]

Just an update: I will end up taking the theoretical course since the physics department decided to change when Quantum Mechanics is being taught and I can no longer take the applied course.
And, no the professors are not the same. I worried because I am planning on also taking algebraic topology (which has point-set topology) as pre req and wanted to focus on physics.

 

[MidgetDwarf]

Can you take the theory class, and then just apply the application of the theory to physics problems?

The Churchill Book is nice, has some proofs in it. But I would supplement it with a book that shows a more geometric approach to Complex Analysis.

 

[mathwonk]

I think actually knowing the content of both courses is useful. So I was going to recommend, since you are taking the theoretical course, that you also audit the applied course to better understand what is going on. But since it seem to pose a scheduling conflict, maybe you could benefit from reading the Churchill book along with your other course. I may be unusually slow, but to try to learn complex, I myself studied (and taught from) Greenleaf, Churchill, Knopp, Ahlfors, Cartan, Lang, and Hille, and browsed in Mackey. I still don't know it all.

Some main points include: the fact that all complex differentiable functions can be represented by power series, the integral formulae, the residue calculus and argument principle, open mapping and maximum modulus principle, removable and essential singularities and poles, conformal mapping, and the identity principle for analytic functions. More advanced are the Riemann mapping theorem, normal families, and the Picard theorems. Churchill is quite good on explicit conformal mappings as I recall (from some 50 years ago).

My favorite to learn from as a beginner, was (Frederick) Greenleaf, and the most beautiful and elegant theoretical book is Henri Cartan. Einar Hille (2 volumes) is the most comprehensive, but may not appeal to everyone. Silverman is very short and elegant, and many people like Markushevich. Knopp is too brief, and Ahlfors was to me somewhat unhelpful. As you may have deduced, there is an unusually large number of good to excellent books on this topic.

 

[wrobel]

I would take the second one:

"Review of analysis in R2 including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, linear fractional transformations; Cauchy’s theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem. "

Especially in regard with physics applications

 

[Jonathanos]

@mathwonk
I appreciate this list, I think it will be very helpful so I can know what to look for specifically. Not that I will only learn this but just to keep in mind while taking the course.[/QUOTE]