Applications of complex analysis?
I am a math major and I really cannot understand complex analysis. I've tried it twice before doing so poorly on the midterms that I had to drop. I gave it a go during this summer and I again ended up dropping it.
I know that all the courses in the curriculum serve some purpose. I've taken differential equations and real analysis, so I can get my degree without ever taking complex analysis, but a lot of people have told me it's an integral part of a maths education, though they never specified why.
I want to know why. Why is complex analysis so important? What area outside of math (besides physics, electromagnetism; cannot stand physics) is complex analysis used?
Thank you
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$\begingroup$Sorry, but when they say this is an integral part of math education they don't mean that you can apply complex analysis to literature or social sciences! I don't know what you're looking for, but complex analysis has millions of applications inside mathematics. Complex numbers and complex analysis show up everywhere in mathematics and physics. Algebraically, complex numbers are closed. This is a good algebraic property for a field. They've been studied in mathematics since the 17th century because of their applications to mathematics, mechanics, waves, etc. Complex numbers show up in number theory a lot. From the analytical point of view, there is a beautiful theory for series in complex analysis. Residues are a powerful tool for computation of integrals. Euler's identity shows us that logarithms, exponential functions and trigonometric/hyperbolic functions and their inverses can be thought of in a unified way. Winding numbers are important objects of study in Algebraic topology, but actually the first ideas of using them came from complex analysis. And many many other reasons that people can say why complex analysis is an important, and also beautiful, branch of mathematics.
$\endgroup$ 2 $\begingroup$Complex analysis is used in 2 major areas in engineering - signal processing and control theory.
In signal processing, complex analysis and fourier analysis go hand in hand in the analysis of signals, and this by itself has tonnes of applications, e.g., in communication systems (your broadband, wifi, satellite communication, image/video/audio compression, signal filtering/repair/reconstruction etc). Basically, if you search for applications of signal processing, those are the applications that are indirectly the applications of complex analysis. Although most engineers will tell you that complex analysis is not necessary to "understand" signal processing, I have found that it is very helpful in going beyond simply blindly applying the fourier transform etc., to a stage where one truly understand what is going on. See this article Connections between signal processing and complex analysis for details.
The second application area is control theory, specifically in the analysis of stability of systems and controller design. Here the word "system" is used generically, and does not necessarily refer to an electrical system. For e.g., one could use it to (try to) understand stock market movement, chemical processes/reactions. Also, control theory is used heavily in robotics, and by extension, so is complex analysis.
I should add that the complex analysis as is taught in the math department is rarely used in its "pure" form in what most people perceive as "real-world applications". For e.g., using complex analysis to help solve abstract-looking equations (e.g., differential equations) that is used to model certain interesting phenomenon (e.g., cellular processes in system biology) is also an application, although one might rarely hear people associate the two directly. Often, engineering applications will only make use of parts of what is taught in a complex analysis course, and usually through another area such as fourier analysis or differential equations. But this does not mean it is "useless".
$\endgroup$ $\begingroup$You learn how to compute real integrals using the residue theorem. This is important and easy, because you don't need strange transformation or hints for computation, you just can relax and computate the residue of a function and sum some of them up.
Outside Mathematics it is unimportant, because it is mathematitcs...
$\endgroup$ 3 $\begingroup$A very direct and beautiful application of complex analysis to the physical world is encapsulated within the Kramers-Kroenig relations. There, the imaginary part of a function is found from its real part, or vice-versa. It is an expression of causality in terms of analyticity. Applications include dispersion relations in optics, as well as others mentioned in the linked article.
$\endgroup$ 1 $\begingroup$I also want to add to previous answers that conformal mappings (one part of the course of complex analysis) are widely used in mathematical physics for solving boundary problems on very complicated domains. As you know, mathematical physics is (perhaps) one of the most fundamental sciences that refer to nature research at the same time with the rest of nature disciplines.
$\endgroup$ 1 $\begingroup$There are some applications in high end fluid dynamics and thermo. (Not sure if you consider that physics, but at least it's not E&M. So MechE and its offshoots: nuclear, aero, etc.
You might consider other areas instead (time is limited) that are more applied or that you enjoy more or are easier for you: advanced linear algebra, PDEs, abstract algebra, etc. Maybe even just programming or CS--always useful.
Perhaps you struggled because you took more of a math approach to the course (lot of rigor, theory, lose the big picture perspective, lose applications motivation). Consider to self study with Stroud or the Schaum's Outline or Silverman's Dover book.
I recommend that you sign up for this Coursera class:
Introduction to Complex Analysis
The instructor is excellent explaining the subject in an intuitive way.
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