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 Excerpt: ... 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 ...Read MoreThis 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 Excerpt: ... 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 thus: 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...Read Less

New. 396 pages. Reprinted from 1902 edition. Smyth/Section Sewn, Hard-Cover. Exact/Strict reproduction of text, no changes has been made in respect to the original text. If the original book was printed in multiple volumes/bindings than this reprint is of only a single volume/binding. A lot of effort has been made to check and improve each page/scan manually for its quality of text and illustrations (if any, are in b/w). Folded illustrations, if any, are not included in the book. This is not a retyped or an ocr'd book. Index, contents, etc, if any in the original book, are included. This item is printed on demand using good quality natural shade paper.

New. 402 pages. Reprinted from 1902 edition. Smyth/Section Sewn, Hard-Cover. Exact/Strict reproduction of text, no changes has been made in respect to the original text. If the original book was printed in multiple volumes/bindings than this reprint is of only a single volume/binding. A lot of effort has been made to check and improve each page/scan manually for its quality of text and illustrations (if any, are in b/w). Folded illustrations, if any, are not included in the book. This is not a retyped or an ocr'd book. Index, contents, etc, if any in the original book, are included. This item is printed on demand using good quality natural shade paper.

New. 572 pages. Reprinted from 1915 edition. Smyth/Section Sewn, Hard-Cover. Exact/Strict reproduction of text, no changes has been made in respect to the original text. If the original book was printed in multiple volumes/bindings than this reprint is of only a single volume/binding. A lot of effort has been made to check and improve each page/scan manually for its quality of text and illustrations (if any, are in b/w). Folded illustrations, if any, are not included in the book. This is not a retyped or an ocr'd book. Index, contents, etc, if any in the original book, are included. This item is printed on demand using good quality natural shade paper.

New. 622 pages. Reprinted from 1920 edition. Smyth/Section Sewn, Hard-Cover. Exact/Strict reproduction of text, no changes has been made in respect to the original text. If the original book was printed in multiple volumes/bindings than this reprint is of only a single volume/binding. A lot of effort has been made to check and improve each page/scan manually for its quality of text and illustrations (if any, are in b/w). Folded illustrations, if any, are not included in the book. This is not a retyped or an ocr'd book. Index, contents, etc, if any in the original book, are included. This item is printed on demand using good quality natural shade paper.