By Jerome Spanier, Keith B. Oldham

This moment variation of An Atlas of services, with Equator, the Atlas functionality Calculator, presents entire details on a number of hundred features or functionality households, of curiosity to all these scientists, engineers and mathematicians who're focused on the quantitative elements in their box. starting with easy integer-valued features, the publication progresses to polynomials, exponential, trigonometric, Bessel, and hypergeometric capabilities, in addition to many extra. The sixty five chapters are prepared approximately so as of accelerating complexity, mathematical sophistication being saved to a minimal whereas application is under pressure all through. as well as offering definitions and straightforward homes for each functionality, every one bankruptcy catalogs extra complicated interrelationships in addition to the derivatives, integrals, Laplace transforms and different features of the functionality. quite a few colour figures in 2 or three dimensions depict their form and qualitative beneficial properties and flesh out the reader’s familiarity with the services. regularly, the bankruptcy concludes with a concise exposition on a subject matter in utilized arithmetic linked to the actual functionality or functionality relatives. beneficial properties that make the Atlas a useful reference device, but easy to exploit, comprise: complete insurance of these functions—elementary and “special”—that meet daily wishes a standardized bankruptcy structure, making it effortless to find wanted info on such features as: nomenclature, normal habit, definitions, intrarelationships, expansions, approximations, limits, and reaction to operations of the calculus wide cross-referencing and finished indexing, with beneficial appendices the inclusion of cutting edge software--Equator, the Atlas functionality Calculator--on a CD with the print publication obviating the necessity for tables or programming to discover numerical values (once put in onto your Windows-based computing device, this precise functionality calculator generates designated functionality values on call for) the inclusion of recent fabric facing attention-grabbing purposes of the various functionality households, development upon the favorable responses to comparable fabric within the first variation.

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Magnus Oberhettinger and Tricomi [Higher Transcendental Functions. Volume 1, pages 38-391. 4:4 SPECIAL CASES When n is an even integer, the Bernoulli number B. is expressible as a zeta number (Chapter 3] 4:4:1 z,n n C(n) B. (n - I - j)! 4.... with Ba = 1. Recursion 4:5:1 is easily derived from the more compact expression 4:5:2 (n"B,=O J where (,) is the binomial coefficient [Chapter 6]. 4.... THE BERNOULLI NUMBERS, B. 37 4:9 4:6 EXPANSIONS Even-indexed Bernoulli numbers are expansible as 2n r B.

2. 3, .... Zeta and lambda functions are expansible as the infinite products 1 3:6:5 6(v) 1 -2 I 3:6:6 1 1 I I 1 I 1 -3 "I -5-" 1 - 7-" 1 I -3-''1 -5-"1 -7-"l - 11- ri 1 7r 1 v>I v>1 where a; is the jth prime number. 202056903 and G is Catalan's constant [see Section 1:7]. For n = 2, 4, 6, ... all zeta, lambda and eta numbers equal e' multiplied by a proper fraction [the fraction is related by equation 3:13:1 to the Bernoulli number B. of Chapter 4]. 1 v=-5 v=-4 -I '=-2 v=-3 I 1 252 120 12 31 -7 1 0 252 120 I -1 0 1 1 4 2 0 6 - 0 v=4 v=s a' Z 90 u' a' 72 8 8 % n' 32 7a' 4 720 In(2) -1 5 v-3 a' 1 2 0 8 0 v-2 v= I 12 0 4 v= 0 0 0 WY) v=-1 12 1 rr 2 d G 0 - rr' 32 A(4) 51 by equation 3:13:2.

A in which S'';' is a Stirling number of the first kind [see Section 18:6). As well, definition 6:3:1 constitutes the expansion of (;,) as a finite product. When v, but not in. is large in magnitude, the asymptotic expansion 6:6:2 vv rr (rn) = - I i M! m(m - 1)(m - 2)(3m - 1) + ... ') 47 6:8 applies. Conversely, when m, but not v, is large, we have 6:6:3 (v \m) ' (-I)" m"' (1(-v) [1+ v(v+ 1Hv +2)(3v+ 1) v(v+ 1) 24m' + 2m + ... m-s 70 where f is the gamma function [Chapter 43]. As noted in Section 6:2, the maximum value of ( ), for a given even value of n.