TITLE: How is complex analysis relevant to other areas of mathematics?
QUESTION [7 upvotes]: I'm going to do a math PhD next year. My main interests in mathematics are composition algebra, algebraic topology, and group theory.
Now I have the opportunity to take a one-semester course in complex analysis, but I am not sure if it is relevant to my research interests.
My understanding of complex analysis is that it is mostly about Laurent series, Cauchy's integral theorem, residue theorem, and evaluating weird-looking real integrals using these techniques. But I fail to see its connection to algebra.
I am sure that it is a beautiful theory and that it is also an important set of tools for applied mathematicians and engineers, but is there anything useful I can take from complex analysis as an algebraist?

REPLY [3 votes]: Your main interests show that you want to become a professional mathematician. In these circles everyone will take it for granted that you know the basics of complex analysis. Computing real integrals using all sorts of residue tricks does not belong to the mainstream of this field.