Algebra, Mathematical Analysis - General & Miscellaneous, Calculus
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Synopsis
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 Excerpt: ... 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 following: 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...Book Details
Published
May 28, 2012
Publisher
Rarebooksclub.com
Pages
58
Format
Hardcover
ISBN
9781231376447