The integration of functions of a single variable

by G.H. Hardy

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. 1916 ... by that of its greatest term. In the exponential series, for example, the greatest term is that for which » = #, and the increase of this term is ejjx. We have assumed /i positive and finite. A slight variation of the argument shows (a) that v = 0 when /i is infinite, and (b) that/(#) is not of finite order when /i = 0. 6.33. Special results. If we make more drastic assumptions about the coefficients an, we can naturally obtain more precise results about/(#). Thus if n(ln)-b... (J»)-6+sr1/,Xa„-« (ln)-b'... (M-6-s-1'", then er"(lr)b'... (lkrf-M(r)erv (lr)b'... (J»r)6+s, and conversely. If where «-'n-XW'n then log/(#)-x (')'. As examples of still more accurate and special results we may quote the nm V ea/' (m!) /av 'r(cm+l) a' where a0 and in the last formula 1 p 2, and #--oo by positive values. These results may of course be used to give an upper limit for the modulus of the particular function considered when x is not necessarily real, and so for Mir). General accounts of the theory of integral functions are given by Borel, 2; Vivanti, 1; Bieberbach, 1; Valiron, 1. The second edition of the first work contains a very valuable note by Valiron on the latest developments of the theory, and the second work a very complete bibliography up to 1906. Particularly important memoirs (beyond those on which Borel's account of the theory is based) are those of Boutroux, 1; Lindelof, 2; Pringsheim, 7; Valiron, 2, 3; and Wiman, 1,2, 3. For more precise and special developments, such as those quoted at the beginning of this section, see in particular Le Roy, 1; Lindelof, 3; Littlewood, 1, 2, 3, 4; and Mellin, 1. For the theory of integral functions of infinite order, see Blumenthal, 1. 6.34. Irregularly increasing functi...

Genres: Mathematics

58 pages, Paperback

Published January 1, 2012

About the author

Godfrey Harold Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.

Non-mathematicians usually know him for A Mathematician's Apology, his essay from 1940 on the aesthetics of mathematics. The apology is often considered one of the best insights into the mind of a working mathematician written for the layman.

His relationship as mentor, from 1914 onwards, of the Indian mathematician Srinivasa Ramanujan has become celebrated. Hardy almost immediately recognized Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan. He called their collaboration "the one romantic incident in my life."