By Keit Oldham, Jan Myland, Jerome Spanier (auth.)
This moment variation of An Atlas of capabilities, 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 facets in their box. starting with easy integer-valued features, the booklet progresses to polynomials, exponential, trigonometric, Bessel, and hypergeometric services, and lots of extra. The sixty five chapters are prepared approximately so as of accelerating complexity, mathematical sophistication being stored to a minimal whereas stressing software all through. as well as delivering definitions and easy houses for each functionality, each one bankruptcy catalogs extra advanced 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 gains and flesh out the reader’s familiarity with the capabilities. generally, the bankruptcy concludes with a concise exposition on a subject matter in utilized arithmetic linked to the actual functionality or functionality family.
Features that make the Atlas a useful reference software, but basic 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 details on such features as: nomenclature, common habit, definitions, intrarelationships, expansions, approximations, limits, and reaction to operations of the calculus
extensive cross-referencing and entire indexing, with beneficial appendices
the inclusion of leading edge software--Equator, the Atlas functionality Calculator
the inclusion of recent fabric facing attention-grabbing purposes of a number of the functionality households, development 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
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!
Note that 3:11:1 is restricted to v > 1, but Figure 3-2 depicts the behavior in a more widespread region of the complex plane. Notice that the discontinuity encountered at v 1 along the real line does not extend beyond that line. Not evident in the figure is that, in accord with Riemann’s hypothesis, there exists a series of complex zeros (that is, points in the complex plane where both the real and the imaginary parts are zero) along the line v ½. THE ZETA NUMBERS AND RELATED FUNCTIONS 36 3:12 3:12 GENERALIZATIONS The four functions of this chapter are special cases of the Hurwitz function of Chapter 64.
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. Here sech is the hyperbolic secant discussed in Chapter 29. An integral definition for Euler numbers of even degree is 5:3:2 En n § 2 · ( ) 2 ¨ ¸ ©S¹ n 1 f n ³ t sech(t )dt n 0, 2, 4, 0 With E0 specified as unity, equation 5:5:2 sometimes serves as a definition of other Euler numbers of even degree.
An Atlas of Functions: with Equator, the Atlas Function Calculator by Keit Oldham, Jan Myland, Jerome Spanier (auth.)