By Keit Oldham, Jan Myland, Jerome Spanier (auth.)
This moment variation of An Atlas of capabilities, with Equator, the Atlas functionality Calculator, offers complete info on a number of hundred services or functionality households of curiosity to scientists, engineers and mathematicians who're concerned about the quantitative points in their box. starting with basic integer-valued capabilities, the publication progresses to polynomials, exponential, trigonometric, Bessel, and hypergeometric features, 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 delivering definitions and straightforward 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. quite a few colour figures in - or 3- dimensions depict their form and qualitative positive factors and flesh out the reader’s familiarity with the features. regularly, 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 instrument, but easy to take advantage of, include:
full insurance of these functions—elementary and "special”—that meet daily needs
a standardized bankruptcy structure, making it effortless to find wanted info on such facets as: nomenclature, normal habit, definitions, intrarelationships, expansions, approximations, limits, and reaction to operations of the calculus
extensive cross-referencing and finished indexing, with necessary 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, development upon the favorable responses to comparable fabric within the first edition.
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Extra info for An Atlas of Functions: with Equator, the Atlas Function Calculator
M ( v ) m m! is possible generally, whereas a definition exclusively in terms of the factorial function [Chapter 2] n! ( n m)! requires that the upper index be a nonnegative integer. §v· THE BINOMIAL COEFFICIENTS ¨© m ¸¹ 6:4 51 §v· ¨0¸ © ¹ §v· ¨1¸ © ¹ §v· ¨ 2¸ © ¹ §v· ¨ 3¸ © ¹ §v· ¨ 4¸ © ¹ §v· ¨5¸ © ¹ §v· ¨6¸ © ¹ 6:4 SPECIAL CASES Reduction to an expression involving the double factorial [Section 2:13] occurs when the upper index is equal to twice the lower index § 2m · ¨ m¸ © ¹ 6:4:1 4 m (2m 1)!!
Round is the rounding function, described in Section 8:13. 2:10 OPERATIONS OF THE CALCULUS No operations of the calculus are possible on a function such as n! that is defined only for discrete arguments. 2:11 COMPLEX ARGUMENT In view of relation 2:12:1, the gamma function formulas given in Section 43:11 may be used to ascribe meaning to (n+im)!. THE FACTORIAL FUNCTION n! 2:12 25 2:12 GENERALIZATIONS The factorial function is a special case of the gamma function [Chapter 43] n! * ( n 1) n 0,1, 2, 2:12:1 and of the Pochhammer polynomial [Chapter 18] n!
N m)! requires that the upper index be a nonnegative integer. §v· THE BINOMIAL COEFFICIENTS ¨© m ¸¹ 6:4 51 §v· ¨0¸ © ¹ §v· ¨1¸ © ¹ §v· ¨ 2¸ © ¹ §v· ¨ 3¸ © ¹ §v· ¨ 4¸ © ¹ §v· ¨5¸ © ¹ §v· ¨6¸ © ¹ 6:4 SPECIAL CASES Reduction to an expression involving the double factorial [Section 2:13] occurs when the upper index is equal to twice the lower index § 2m · ¨ m¸ © ¹ 6:4:1 4 m (2m 1)!! (2m)!! (2m)! ) 2 m 0,1,2, or differs by unity from twice the lower index 6:4:2 § 2m 1 · § 2m 1 · ¨ m 1 ¸ ¨ m ¸ © ¹ © ¹ 2 2 m 1 (2m 1)!!