568 Chapter 9 • Sequences and Series of Functions00 1- Riemann's Zeta function^8 is defined ((x) = I:; kx, for all x > 1.
k=l
Prove that this series converges uniformly on the interval [a, oo) for any
(^00) Ink
a> 1, and show that \:/x > 1, ((x) is differentiable and ('(x) = - k'fl kx.
9.4 *Two Results of Weierstrass
This section is designed as another project to be completed by
students. It has no exercise set at the end. Instead, students
are asked to complete the proofs following guidelines provided
in the text.
Karl Theodor Wilhelm Weierstrass (1815- 1897), often called "the father of
modern analysis," was a late bloomer. At his father's insistence he enrolled at
the University of Bonn in 1834 to study law, finance, and economics instead
of mathematics, the subject of his real interest. He left there after four years
of intensive fencing and drinking, without having studied the required courses
and without taking the final examinations. Continuing to study mathematics
privately, he resolved to become a mathematician and enrolled at the Academy
in Munster in 1839 to become a secondary school teacher. There his genius
attracted the attention of Professor Gudermann, who gave him strong encour-
agement. While a secondary teacher for some 15 years, Weierstrass continued
mathematical research and wrote several papers on elliptic functions. His 1854
paper on Abelian functions brought him widespread recognition, an honorary
doctoral degree, and an offer of appointment at any university of his choice
in Austria. He chose to wait for an appointment at the University of Berlin,
which he received in 1856. There he lectured on analytic functions (complex
variables), elliptic functions (his main area of research), Fourier series and in-
tegrals, calculus of variations, and applications to geometry and mathematical
physics. His lectures attracted students from all over the world. His insistence
on rigorous standards of proof is still a dominating force in the teaching of
analysis today.
- Riemann's Zeta function has important applications in physics and m athematics. It is used
in number theory in the study of the distribution of prime numbers.