A course of modern analysis; An introduction to the general theory of infinite series and of analytic functions, with an account of the principal transcendental functions

Cambridge Mathematical Library

by Edmund Taylor Whittaker

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 ... this from the Fourier expansion we should have an expansion (oo-c„)+ 2 (am-cm) cos mz + (bm-dm) sin mz m=l whose sum is zero for all values of z between 0 and 2ir, except possibly a certain finite number of values (namely the discontinuities). The investigation therefore turns on the question whether it is possible for such an expansion as this last to exist. We shall shew that it cannot exist, and that consequently the Fourier expansion is unique. Let" Aa = a„-4 m = am cos mz + bm sin mz (m 1); and let t = A0 + A, +... + Am +... be a convergent (not necessarily absolutely convergent) series for values of z from 0 to 2tt, So that the limit of an and bn is zero for n = oo; and suppose that (except at certain exceptional points) its sum is zero. Then the series F()=a0;-aa-...--... converges absolutely and uniformly for this range of values of z, as is seen by comparing it with the series S. We shall first establish a lemma due to Riemannf, which may be stated The proof is due to G. Cantor, Journal filr Math, Lxxii. + Collected Workt, p. 213. As F(z) converges absolutely, we can rearrange the order of the terms, and so can write n.. /sin a. /sin 2as R = A0 + A1-)+A2-)+.... Now considering the series S, we can write A„ + A, + At+... + An_1=f(z) + en, say, where z being given, and any small quantity 8 being assigned at will, we shall have en 5 for values of n some integer m. Now An = en+i--e„ for all values of n. Therefore substituting, we have n=l U (n!)« i I «« J. Divide the series on the right-hand side of this equation into three parts, for which respectively (1) lnm, (2) m + 1 n s, where s is the greatest integer in-, (3) s + ln. The first part consists of a finite number of terms, each tending to zero as a...

Genres: Mathematics, Science

64 pages, Paperback

First published January 1, 1927

About the author

1873-

Reviews

[Wissam Raji · Rating 5 out of 5]

A must-have classic in the theory of classical analysis. Treatment of topics is highly beneficial for number theorists, analysts, and algebraists. It is one of the excellent references to own. The topics treated about the special functions are rare to find in other books and are very useful for students in analytic number theory.

[The Architect]

For an up-to-date version of the treatment of special functions, see the "Bateman Manuscript Project".