By Jerome Spanier, Keith B. Oldham

This moment variation of An Atlas of services, with Equator, the Atlas functionality Calculator, presents complete info on a number of hundred services or functionality households, of curiosity to all these scientists, engineers and mathematicians who're taken with the quantitative points 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 stored to a minimal whereas software is under pressure all through. as well as offering definitions and easy homes for each functionality, each one bankruptcy catalogs extra advanced 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 positive factors and flesh out the reader’s familiarity with the features. repeatedly, the bankruptcy concludes with a concise exposition on an issue in utilized arithmetic linked to the actual functionality or functionality kin. beneficial properties that make the Atlas a useful reference instrument, but basic to exploit, comprise: complete assurance of these functions—elementary and “special”—that meet daily wishes a standardized bankruptcy structure, making it effortless to find wanted details on such facets as: nomenclature, common habit, definitions, intrarelationships, expansions, approximations, limits, and reaction to operations of the calculus broad cross-referencing and accomplished indexing, with priceless appendices the inclusion of leading edge software--Equator, the Atlas functionality Calculator--on a CD with the print e-book obviating the necessity for tables or programming to discover numerical values (once put in onto your Windows-based laptop, this detailed functionality calculator generates special functionality values on call for) the inclusion of recent fabric facing attention-grabbing functions of the various functionality households, development upon the favorable responses to comparable fabric within the first version.

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5:5 INTRARELATIONSHIPS The compact expression 5:5:1 (2n 2J/ =0 n= 1,2,3.... where (;) is the binomial coefficient treated in Chapter 6, gives rise to the recursion formula for even-indexed Euler numbers 5:5:2 with Ev = 1. E,, E"=-n! (2j)! 6, n J=--1 2 THE EULER NUMBERS, E. ( 5:6:1 l" } rr t (-I)1 1)"/2_nii L1-3, i+5 R+ n=0,2,4,... 5:7 PARTICULAR VALUES E. E, 1 0 F. 0 0 5 Eb E, F. E. E. E E E13 Eu -61 0 1385 0 -50521 0 2702765 0 -199360981 I I E,, E,e 0 19391512145 5:8 NUMERICAL VALUES For n = 2, 4, 6, ...

3:4 SPECIAL CASES The zeta, lambda, eta and beta numbers may be regarded as special cases of the corresponding functions. No further specialization is fruitful. 3:5 INTRARELATIONSHIPS The zeta and beta functions satisfy the reflection formulas (1 - v) = 3:5:1 2r(v)4(v) (2a)" V'Tr cost 2 p(I - v) = I -J r(v)p(v) sine 3:5:2 involving the gamma function [Chapter 43) and the functions of Chapter 32. With f(n) representing any one of the four numbers I,(n), Mn). 11(n) or (3(n). one may sum the infinite series E(-1)"f(n)/n, as well as the series of complements E[I - f(n)], E(-1)"(1 - f(n)].

4:3 DEFINITIONS The Bernoulli numbers are defined through the generating function 4:3:1 x _ exp(x) - l - e B" n! m THE BERNOULLI NUMBERS, B. 0 6 0 FIG 4-1 B, -I B. 30 0 Bernoulli ' number : -B: O. S B, B. B. B, B; .............. 5 30 Rounded value B 8 l0-: -5 x 10 g -2 x 10' B. B. B. * B, 6 x 10' 8x10" 3 x 104 x 10'° The integral representations of the even-indexed Bernoulli numbers include 4:3:2 B. _ (-1)'"''' -sr Jv t"csch'(t)dr n = 2, 4, 6, ... where csch denotes the hyperbolic cosecant function [Chapter 291.