Applications to Other Areas of Mathematics

We have already seen, especially in Chapter 11, howthe methods of complex analysis can be applied to the solution of problems from other area of mathematics. In this chapter we will get some insight into the fantastic breadth of such applications. For that reason, the topics chosen are rather disparate. Section 19.1 involves calculating the total variation of a real function, and illustraties how the methods of Chapter 11 can be applied to yet another nontypical problem. Section 19.2 offers a proof of the classic Fourier Uniqueness Theorem using two preliminary results from real analysis and a surprising application of Liouville's theorem. In Section 19.3 we see how the use of a generating function allows complex analytic results to be applied to an infinite system of (real) equations. Generating functions are also the key to the four different problems in number theory that comprise section 19.4. Finally, in section 19.5, we offer a well-trimmed analytic proof of the prime number theorem based on properties of the Zeta function and another Dirichlet series.