# How difficult is complex variables?

• Courses
• ispivack
In summary, Complex analysis is not too difficult, but it does require a good understanding of real world variables. If you want to take another course, I think it is reasonable based on your current load.

#### ispivack

Hello, I am a rising sophomore in Astronomy and Physics. I am taking complex variables next semester and was wondering the effort required to succeed in the class. There are some other classes I'd like to take, however I don't want to overload myself. I have taken up through multivariable calc, diff eq, and linear algebra at this point. How does complex variables compare to those courses as far as time required goes? As a side note, other than complex variables I am taking Thermo/Waves, computational physics, and an intro to proofs class.

It is not very hard, so don't worry about it.

The fact that the variables are complex isn't very difficult, as they are still variables. The difficulties come from the fact, that we have a far better understanding of real variables, so many calculations are reduced to the real components, and here is where complexity starts. ##\mathbb{C} \neq \mathbb{R}^2## as a field. The complex numbers have a richer structure than their real components combined. You can see this even for some basic functions as roots, logarithms or trig functions when applied to complex numbers. Strange things happen to complex valued functions. E.g. differentiability isn't the differentiability of its real parts, an additional condition is needed. So the enrichment comes to a prize - or a benefit, depending on how you want to look at it.

But isn't the question here a complete different one? What is the alternative? And I don't mean alternative courses, I mean that there is no way around complex analysis in physics. It is simply a requirement you can't avoid. Therefore the question is now or later, not now or never.

And yes, @Dr.D is right. It's not very difficult. As an advice: do not compare them with the real case. It's easier if you consider it as new.

johnnyringo6
fresh_42 said:
But isn't the question here a complete different one? What is the alternative? And I don't mean alternative courses, I mean that there is no way around complex analysis in physics. It is simply a requirement you can't avoid. Therefore the question is now or later, not now or never.

The alternative would be to take another class as well. I have many broad interests and would like to take more culture/language classes before grad school. I was just trying to figure out if it would be reasonable to take another course based on my current load. I think I will enroll in it, then drop it if the workload is too heavy.

When I took complex analysis it was taught in the math department and was very "proofy." I lost track of what we were doing, and why we cared about it. Eventually some of the methods appeared in my physics courses and it started to make sense. I should probably have gone back and re-studied the proofy stuff, I would have a better appreciation for it.

When I took Complex Analysis, it was not proof-based, nor was it very algebraic. My professor spent the entire semester focusing on the geometric meaning of operations on complex numbers. Like, what does it mean to multiply two complex numbers (the rotation and dilation of their vector representations), or what does it mean to differentiate/integrate in the complex plane (the "amplitwist", etc.)? It was VERY helpful when we got to Laurent Series and the Residue Theorem, because we were able to understand almost immediately why and how those concepts worked. My only complaint was the lack of algebraic manipulation, because I still struggled with that on the final exam, due to lack of practice (which was partly my fault).

All in all, I'd highly recommend taking the course, but spend some time asking yourself and the professor what it means, both geometrically and algebraically, to do all of these operations. I think it takes a sufficient understanding of both to be truly successful in the course.

## 1. How important is it to understand complex variables in science?

Complex variables are essential in many areas of science, including physics, engineering, and mathematics. They provide a powerful tool for describing and analyzing systems with multiple variables and are necessary for understanding many phenomena in the natural world.

## 2. Is complex variables difficult to learn?

Complex variables can be challenging to learn, as they involve abstract concepts and require a strong foundation in algebra and calculus. However, with dedication and practice, it is possible to gain a solid understanding of complex variables.

## 3. Are there any real-world applications of complex variables?

Yes, there are numerous real-world applications of complex variables, including electrical engineering, fluid dynamics, quantum mechanics, and signal processing. They are also used in fields such as economics, statistics, and computer science.

## 4. How does understanding complex variables benefit scientific research?

Understanding complex variables allows scientists to model and analyze complex systems with multiple variables, which is crucial for making accurate predictions and developing new theories. It also enables the use of advanced mathematical techniques in various fields of research.

## 5. What resources are available for learning complex variables?

There are many resources available for learning complex variables, including textbooks, online courses, and tutorials. Additionally, attending workshops and seeking guidance from experienced scientists can also be helpful in mastering this subject.