To Take Complex Analysis or Not?

[cwatki14]

So I will be a sophomore this next semester, and I am having difficulty deciding whether or not to take complex analysis. I am majoring in chemical and biomolecular engineering (with a concentration in cellular/molecular engineering), but I feel after this past semester my heart really lies with math.

Number theory was my first proof-based class, and seeing as I had no knowledge as to how to write a proof it kicked me around for the first half of the semester. Nonetheless, I really enjoyed the class. My professor for number theory suggested I take Advanced Algebra I next, since the courses apparently overlap a good deal. However, there is a time conflict with Biochemistry, which I need for the ChemE degree. He then suggested if I take the analysis route to take complex analysis before real.

So my question is how is complex analysis in comparison to the math courses I have already taken (single variable calculus, multivariable calculus, linear algebra, and number theory.) I heard it can be a bit tricky at first, but apparently the book used for the course is excellent. Next semester I am taking thermodynamics, biochemistry, differential equations, and a course called the scientific revolution. My only concern is that complex analysis may push me to the point of insanity.

 

[niklaus]

Of course it all depends on how your professor will do the class, but I would expect that he/she will take many concepts from real analysis for granted and focus on what changes when you go from real variables to complex ones. It might involve quite a bit of extra work on your part to catch up on the ideas real and complex analysis have in common. Maybe you should check with the professor if it's a good idea to do complex before real...

 

[Llewlyn]

I think that the difficulty of the subject strictly depends on the topics covered. Complex analysis usually involves the study of the so-called holomorphic functions, by techniques which are nothing more than multivariable calculus. If you instead cover more advanced topics such as Riemann's surfaces it can be tricky.

Ll.

 

[cwatki14]

At my university real and complex variables are separate courses and are also distinct from real and complex analysis. Also, I have heard if you can do multivariable calculus, complex analysis should not be much harder, though they are distinct fields.
The course description reads:
"This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions."

 

[Shackleford]

At my university real and complex variables are separate courses and are also distinct from real and complex analysis. Also, I have heard if you can do multivariable calculus, complex analysis should not be much harder, though they are distinct fields.
The course description reads:
"This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions."

Yeah, we have an intro to complex analysis and intro to real analysis. The latter is a very tough course from what I hear.

 

[Klockan3]

At my university real and complex variables are separate courses and are also distinct from real and complex analysis. Also, I have heard if you can do multivariable calculus, complex analysis should not be much harder, though they are distinct fields.
The course description reads:
"This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions."

So this is basically a complex calculus course, they often call that complex analysis. It is funny since that means that real analysis is a higher level course than complex analysis :p

Anyway, I don't see at all why that course load would be killing you, it seems pretty normal to me.

 

[MathematicalPhysicist]

So this is basically a complex calculus course, they often call that complex analysis. It is funny since that means that real analysis is a higher level course than complex analysis :p

Anyway, I don't see at all why that course load would be killing you, it seems pretty normal to me.

Normality is so relative nowadays, your normal situation might kill someone else.

 

[cwatki14]

Normality is so relative nowadays, your normal situation might kill someone else.

Biochemistry and thermodynamics for biomolecular engineers are two very brutal courses for my major, which both require a lot of time for studying. It's really a matter of how much time do I have. Last semester, I took 12 credits of chemistry and chemical engineering courses and eight credits of math. Let's just say it wasn't fun when I was in the library until 4 am multiple days each week working on problem sets. I just don't want that to happen again.

And yes, real analysis is rated at a 400 lvl course, while methods in complex analysis is a 300. Granted, sometimes these numbers have no indication of difficulty.

 

[Shackleford]

Here's my university's courses:

MATH 3364: Introduction to Complex Analysis
Cr. 3. (3-0). Prerequisite: MATH 3331. The complex number system, analytic functions, the Cauchy integral theorem, series representation, residue theory, and conformal mapping.

4331;4332: Introduction to Real Analysis
Cr. 3 per semester. (3-0). Prerequisite: MATH 3334 or consent of instructor. Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals.

One is a junior-level course, and the other is a senior sequence.

 

[Tedjn]

I'm very happy that you've become interested in math, because I have a fondness for the subject. Don't discount the advice of your professor; he likely has more knowledge about Hopkins's math program than any of us. At my university, real analysis is generally taken before complex analysis, but neither is fully dependent on the other.

 

[zpconn]

If this is just an undergraduate course in complex analysis, I think you should be totally fine doing it before real analysis. Such a course shouldn't be too hard, and the topics should be exceptionally pretty. :D

In my experience it's the graduate-level introduction to complex analysis that requires a prior familiarity with analysis, as it will go at a much more rapid pace, develop ideas in greater generality and precision, the exercises will be entirely proof-based, etc. In an ordinary undergraduate course the exercises will probably be mostly computations with a few simple proofs.

 

[PieceOfPi]

I agree with zpconn. I just took an undergraduate course complex analysis this term, and it was a pretty straight forward class. Exercises with mostly computations with some proofs. You will see some epsilon-delta type of proof while proving theorems, but I think your class will talk a little bit about this (if not, you can probably learn this on your own).

Although I said it is straight forward, that doesn't mean it is boring. In fact, there are a lot of beautiful results when you get to learn about the integration theory. Wanna prove Fundamental Theorem of Algebra? Use Liouville's Theorem, and done. Wanna take an improper integral of a real function that looks nasty when you integrate by using techniques from Calc II? Use theory of Residues, and you'll find it much easier that way. There are a lot of fascinating results in complex analysis that I cannot summarize all of them on here, but if you enjoy studying math, there is a very good chance that you will enjoy studying complex analysis too!

 

[tenparsecs]

What zpconn and POP said is accurate. Undergraduate complex analysis does not really need any of the ideas from real analysis. It's also a much easier course than real analysis.

 

[Llewlyn]

Here's my university's courses:

It seems to me a very introduction on the most fundamental aspect. If your skill in calculus is solid I think you could handle them.

Ll.

 

[Shackleford]

It seems to me a very introduction on the most fundamental aspect. If your skill in calculus is solid I think you could handle them.

Ll.

Well, that's good to hear. That Intro to Complex Analysis course is a prerequisite to the graduate Methods of Mathematical Physics course at my university, where I'll probably be going to graduate school unless I can get into a much better Tier 1 university. The graduate program allows you to take at most three undergraduate courses, so I'll wait until then to take it.

 

[cwatki14]

Well, that's good to hear. That Intro to Complex Analysis course is a prerequisite to the graduate Methods of Mathematical Physics course at my university, where I'll probably be going to graduate school unless I can get into a much better Tier 1 university. The graduate program allows you to take at most three undergraduate courses, so I'll wait until then to take it.

The only two required courses for the major are Advanced Algebra I and Analysis I. Beyond that it is just about filling distribution requirements. So I need to take 1 more course in analysis (beyond Analysis 1), and I have already fulfilled my other upper level Algebra requirement. Ideally, I would have taken Advanced Algebra 1 next semester, but I can't due to schedule conflict. Sadly, my engineering degree takes precedent over math. All of the other Analysis courses pretty much require Analysis 1 as a prereq. Analysis 1 also does not fit into my schedule, and either way it is taught by the most horrid postdoc next semester. So I think I am going to sit in on Complex Analysis, and hopefully it works out that I stay in it. I guess I'll take Algebra 1 in spring. This summer I am just going to read some books that will help me learn the methods of writing proofs more in depth so I don't fall behind in that aspect.