By Keit Oldham, Jan Myland, Jerome Spanier (auth.)
This moment variation of An Atlas of features, with Equator, the Atlas functionality Calculator, offers entire details on numerous hundred capabilities or functionality households of curiosity to scientists, engineers and mathematicians who're all in favour of the quantitative features in their box. starting with easy integer-valued capabilities, the publication progresses to polynomials, exponential, trigonometric, Bessel, and hypergeometric capabilities, and plenty of extra. The sixty five chapters are prepared approximately so as of accelerating complexity, mathematical sophistication being saved to a minimal whereas stressing software all through. as well as offering definitions and easy homes for each functionality, each one bankruptcy catalogs extra complicated interrelationships in addition to the derivatives, integrals, Laplace transforms and different features of the functionality. a variety of colour figures in - or 3- dimensions depict their form and qualitative positive aspects and flesh out the reader’s familiarity with the capabilities. oftentimes, the bankruptcy concludes with a concise exposition on an issue in utilized arithmetic linked to the actual functionality or functionality family.
Features that make the Atlas a useful reference device, but easy to take advantage of, include:
full insurance of these functions—elementary and "special”—that meet daily needs
a standardized bankruptcy layout, making it effortless to find wanted info on such points as: nomenclature, normal habit, definitions, intrarelationships, expansions, approximations, limits, and reaction to operations of the calculus
extensive cross-referencing and complete indexing, with worthwhile appendices
the inclusion of leading edge software--Equator, the Atlas functionality Calculator
the inclusion of latest fabric facing fascinating purposes of some of the functionality households, construction upon the favorable responses to related fabric within the first edition.
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Additional resources for An Atlas of Functions: with Equator, the Atlas Function Calculator
2 1 1! 2 f 1 2! 2 1 ¦ j ! 2795 85302 33607 j 0 where I0 is the modified Bessel function [Chapter 49]. The corresponding series with alternating signs sum similarly to exp(1) and to the particular value J0(2) of the zero-order Bessel function [Chapter 52]. There is even the intriguing asymptotic result [see equation 37:13:4] f f j 0 0 ¦ ( ) j j ! ~ ³ 0! 1! 2! 59634 73623 23194 Moreover, the series (1)n/(2n)! sums to cos(1) and there are several analogous summations. 2:6 EXPANSIONS Stirling’s formula [see also Section 43:6] 1 1 139 ª º n!
1)!! 0!! 1!! 2!! 3!! 4!! 5!! 6!! 7!! 1 1 1 2 3 8 15 48 105 384 945 3840 10395 46080 135135 645120 8!! 9!! 10!! 11!! 12!! 13!! 14!! THE FACTORIAL FUNCTION n! 26 2:14 Note that, apart from 0!! 1, the double factorial n!! shares the parity of n. Also notice that, to accord with the n 1 instance of the general recursion formula ( n 2)!! ( n 2) n !! 2:13:6 (1)!! is assigned the value of unity. With a similar rationale, one sometimes encounters the values (3)!! 1, (5)!! 1 3 , etc. /n!! of the double factorials of consecutive integers.
Note the very rapid increase in the absolute value |En| with increasing even n, which dictated the logarithmic presentation in the figure. B. 1007/978-0-387-48807-3_6, © Springer Science+Business Media, LLC 2009 45 THE EULER NUMBERS En 46 §E · 5 frac ¨ n ¸ © 10 ¹ 10 5:2:2 n 5:3 4,8,12, where frac denotes the fractional-value function [Chapter 8]. 5:3 DEFINITIONS The generating function f sech(t ) 5:3:1 ¦E n 0 tn n n! may be used to define the Euler numbers, though the summation is slow to converge for larger t values.