Applications of complex analysis?

Question

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

Answer 1

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...

Answer 2

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.

Answer 3

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.

Answer 4

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.