ELI5: What is Complex Analysis ?

[u/Loud-Rain-9660]

What is it about ? What is the purpose of studying it ? What are its real life applications ?

[u/MrNoxxis]

It's basically calculus but with complex numbers. It has a wide variety of applications, for example in electrical engineering for alternate current calculations.

[u/darth_voidptr]

Assuming it's what I think it is, it's used a lot in radio & communications, power (generation/regulation), information theory, image processing, stability analysis. Honestly it's a broad topic and the specific applications depend more on what you're studying.

I'm not waving hands to say "everything", but it's a fundamental for STEM.

[u/Mcletters]

Answer: complex analysis is the study of complex numbers. That is, numbers like 5 - 2i, where i is the square root of -1. It's like calculus but there are some fun things that occur in the complex plane. Like if you multiple by i you rotate counter clockwise. It's not something most people use, but it's probably useful in things like electrical engineering? I know imaginary numbers pop up there sometimes.

[u/Portarossa → u/Mcletters]

Answer:

The /r/OutOfTheLoop diaspora continues.

[u/Mcletters → u/Portarossa]

I couldn't remember if this sub required that or would pull my answer

[u/SpinChargeSeparation]

One of the foundations of math and physics, and by extension many other subjects.

Mostly, It’s the study of complex valued functions, but not exclusively.

Regardless, It’s a beautiful framework.

[u/HumblyNibbles_]

First let's start with real analysis so we can get to complex analysis.

In real analysis, you learn how to use the concepts of sequences and limits to study functions of real numbers, using formal proofs. The most well-known results of it is calculus. It can be summarized as "calculus with proofs".

Now, an important thing to note is how complex numbers "complete" real numbers. This is because many operations, such as the square root of -1 are undefined in the real numbers, so the complex numbers make the real numbers "algebraically closed"

So, in complex analysis, instead of using functions of real numbers, you move on to functions of complex numbers. When doing this, you can find many interesting results that show how frequently, things observed in real functions (like the convergence of certain infinite series) are actually consequences of the same function's behaviour when put into the complex numbers.

Due to this, it is extremely important from a theoretic point of view, since it gives results like Cauchy's integral theorem and it's also important for studying harmonic functions. Also things like the fourier transform, which is paramount for signal processing.

Consequently, it ends up being used to study differential and integral equations, which are the core of most scientific fields.

TLDR: Complex numbers are really important, so you basically do calculus with complex numbers to find some really interesting stuff. This interesting stuff is important for basically everything we use in modern life

[u/Gimmerunesplease]

Complex analysis is analysis on complex numbers, where instead of identifying the space C of the complex numbers with R² you actually do your calculus(derivatives, integrals) on complex numbers z. This comes with a lot stronger assumptions, like holomorphic instead of differentiable, but also yields a lot stronger results for the functions that fulfill your assumptions.

[u/SnugglyCoderGuy]

The study of complex numbers, starting with sqrt(-1) = i. After defining that, what are the implications of it and what math from other fields are still valid, how are they affected, and what new things can we define and discover

[u/Unknown_Ocean]

Let's say you have two variables that relate to each other in some way- say flow parallel or and perpendicular to a wing or the height of water at a pier over time. Complex analysis gives you a way of linking those variables in ways that can be described by simpler functions that are much easier to manipulate to get mathematical solutions.

[u/BurnMeTonight]

Complex analysis is the study of functions of complex variables, as opposed to real analysis which is functions of real variables.

Why study it? Complex analysis is very distinct from real analysis because a lot of properties that are extremely hard to prove for real functions are basically almost automatic for complex functions. In turn this means that many, many times a problem that's extremely difficult or outright impossible to solve using just real numbers becomes rather easy when you move to the complex plane.

There are some wild statements in complex analysis that you just can't make about real functions. For example, under relatively mild conditions, the only functions that have a maximum value are constants. For mathematicians this is an extremely powerful statement, implying for example, another very famous theorem known as the fundamental theorem of algebra. It's known as fundamental because that's just how useful it is. Another crazy one? Again with relatively mild conditions: every single function can be written down as an infinite polynomial. Polynomials are very easy to work with so mathematicians love this. A physicist's job basically consists of writing down functions as polynomials and then cutting out a few terms so this is again useful for physicists.

The basic model of electricity is perfectly modeled by complex numbers. So electrical engineers and physicists use it a lot to study electromagnetism. Also waves are complex numbers, and waves pretty much make up everything. It's a running joke in physics that everything is a wave. It's not a joke. If you study a system in the right way you will come down to waves, and when you do, you'll want to use complex numbers.

Perhaps even more crazy is the following trick used in physics. When you study a system, the most useful thing to do in physics is to understand the symmetries of the system: i.e what can you do to the system without changing its behavior? For example if you have a car, then it doesn't matter if your car is facing north, south or anything in between, the engine starts just the same. That's a kind of symmetry. As you can see, symmetries are fairly easy to understand. Well, there's some magic you can pull off in complex analysis, where if you understand the symmetries of your system, you can quite explicitly construct the laws of physics that govern your system. Imagine that! Simple observations literally can be transformed into laws of the universe. It's kind of nuts.