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Topic: Handbook of Mathematical Functions

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	A Short History of Mathematical Function Tables
		Functions describing solutions to these problems were often expressed as infinite series, as integrals, or as solutions to differential equations.
		Functions that cropped up most frequently in scientific calculations were given names and notations which have come into common usage: Bessel functions, Struve functions, Mathieu functions, the spherical harmonics, the Gamma function, the Beta function, Jacobi functions, and most of the others appearing on this website.
		Parallel to the handbooks dealing with series expansions, differential equations, functional identities, and so forth of special functions, many integral tables were developed in the 20th century.
	functions.wolfram.com /About/history.html (991 words)

	Physics Today April 2001
		Some of these functions were (with one class of exceptions) known to mathematicians in 1964, but they were not well known to scientists, and had rarely been applied in physics.
		Functions of hypergeometric type can be ordered by the behavior of singular points of the differential equations representing them, or by a group-theoretical analysis of their symmetries.
		Of the two Airy functions, Ai is the one that decays towards infinity, while Bi grows; the J Bessel functions are regular at the origin, the Y Bessel functions have a pole or a branch point.
	www.physicstoday.org /pt/vol-54/iss-4/p11.html (1453 words)

	C++ Mathematical Special Functions Proposal
		While these functions are clearly numerical in nature and will likely be most heavily used by the scientific and engineering communities, other communities of programmers also have needs, ranging from frequent to intermittent, for these functions.
		This Special Functions proposal additionally falls into the "Standards Coordination and Infrastructure" categories identified in [Austern] as targets for the TR, for this proposal is based on an existing standard, Quantities and units [ISO:31] Part 11: Mathematical signs and symbols for use in the physical sciences and technology.
		All the functions are mathematically well-understood, all have proven their utility in practice over a considerable period of time, and all have been previously implemented in C and C++.
	www.open-std.org /jtc1/sc22/wg21/docs/papers/2003/n1422.html (4758 words)

	A New Testament for Special Functions?
		So far, theologicians have steered clear of mathematical alphas and omegas (not to mention the gammas and zetas), but that may be only for lack of access to in-formation about the remarkable properties of special functions.
		Almost any special function is actually part of a family of functions (the constant function 1 is a member of many, many families), but some, such as the gamma function and the Riemann zeta function, stand alone.
		One characteristic of these functions is a tendency to appear in a variety of mathematical and physical circumstances.
	www.siam.org /siamnews/03-98/function.htm (1534 words)

	[No title]
		Be careful though that the recurrence relations for this modified Hankel's functions are somewhat different from those of the normal Bessel and Hankel's functions.
		Reference: M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, section 9.9, and A. Erdelyi, Higher Transcendental Functions, section 7.2.3.
		Function: AiryAi, AiryBi - The Airy wave functions Calling Sequence: AiryAi(x) AiryBi(x) AiryAi(n,x) AiryBi(n,x) Parameters: x - an algebraic expression n - an algebraic expression, assumed to be a non-negative integer Description: - The Airy wave functions AiryAi and AiryBi are linearly independent solutions for w in the equation w''-x*w=0.
	www.math.niu.edu /~rusin/known-math/99/mod_bessel (711 words)

	Shapiro Science Library - Mathematics Subject Guide
		We strive to collect all significant mathematical works in a comprehensive manner in the following areas of mathematics: mathematical analysis, number theory, mathematical statistics and probability, advanced algebra including group theory, modern geometry, algebraic geometry and topology, and the history of mathematics.
		Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, volume 55 of National Bureau of Standards Applied Mathematics Series by M. Abramowitz and I. Stegun.
		Mathematics Archives, maintained on a server at the University of Tennessee Knoxville mathematics department, a rich source for links to web based mathematics resources.
	www.lib.umich.edu /science/math (1423 words)

	AMERICAN MATHEMATICAL MONTHLY -August-September 2006
		The Handbook of Mathematical Functions is one of the most widely circulated mathematical references.
		It is also one of the few large, collaborative mathematical projects and a rare example of a twentieth-century mathematical activity led by a woman.
		The focus of the article is a gentle introduction to the lattice itself in three of its many guises: as a way to encode parking functions, as a key part of the foundations of noncommutative probability, and as a building block for a contractible space acted on by a braid group.
	www.maa.org /pubs/monthly_aug_sep06_toc.html (447 words)

	135-139
		The Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [1] was the culmination of a quarter century of NBS work on core mathematical tools.
		Evaluating commonly occurring mathematical functions has been a fundamental need as long as mathematics has been applied to the solution of practical problems.
		In spite of the fact that sophisticated numerical methods have been embodied in well- designed commercial software for many functions, there continues to be a need for a compendium of information on the properties of mathematical functions.
	nvl.nist.gov /pub/nistpubs/sp958-lide/html/135-139.html (2156 words)

	Cumulative Distribution Functions, Inverses, Random variables (Site not responding. Last check: 2007-11-03)
		Formula 26.6.2 of Abramowitz and Stegun, Handbook of Mathematical Functions (1966) is used to reduce the computation of the cumulative distribution function for the F variate to that of an incomplete beta.
		Formula 26.5.26 of Abramowitz and Stegun, Handbook of Mathematical Functions (1966) is used to reduce calculation of the cumulative distribution function to that of an incomplete beta.
		Formula 26.4.21 of Abramowitz and Stegun, Handbook of Mathematical Functions (1966) is used to reduce the computation of the cumulative distribution function to that of computing a chi-square, hence an incomplete gamma function.
	pauillac.inria.fr /cdrom_a_graver/www/scilab/doc/manual/docu017.html (3539 words)

	Confluent Hypergeometric M Function
		The Handbook of Mathematical Functions (see the REFERENCE section below) defines 2 confluent hypergeometric functions: M and U. This command computes the M function.
		The integral representation of the confluent hypergeometric M function is
		Many common functions can be defined as special cases of this function (see table 13.6 in the Handbook of Mathematical Functions).
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/chm.htm (231 words)

	M. Abramowitz and I. A. Stegun. Handbook of mathematical functions
		The record of continuing acceptance of the Handbook, the praise that has come from all quarters, and the fact that it is one of the most-quoted scientific publications in recent years are evidence that the hope expressed by Dr. Astin in his Preface is being amply fulfilled.
		The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. Miller, and L. Rosenhead.
		The maximum end-figure error, or "tolerance" in the tables in this Handbook is 6/10 of 1 unit everywhere in the case of the elementary, functions, and 1 unit in the case of the higher functions except in a few cases where it has been permitted to rise to 2 units.
	mintaka.sdsu.edu /faculty/wfw/ABRAMOWITZ-STEGUN/intro.htm (2412 words)

	bessely (MATLAB Functions)
		If one input is a row vector and the other is a column vector, the result is a two-dimensional table of function values.
		The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind,
		function uses a Fortran MEX-file to call a library developed by D. E Amos [3] [4].
	www-rohan.sdsu.edu /doc/matlab/techdoc/ref/bessely.html (288 words)

	Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables
		set covers mathematics and the physical world, mathematics and social science, and the laws of chance, with nontechnical essays by and about scores of eminent mathematicians, economists, scientists, and others.
		set includes a general survey of mathematics; historical and biographical information on prominent mathematicians throughout history; material on arithmetic, numbers and the art of counting, and the mathematics of space and motion.
		A distinguished mathematician considers problems in several fields of mathematics, including set theory, algebra, metric and topological spaces, topological groups, analysis, physical systems, and the use of computers as a heuristic aid.
	store.doverpublications.com /0486612724.html (263 words)

	33: Special functions
		After the fact we have learned: (a)about Bessel functions (2)to use a symbolic algebra program to solve differential equations.
		Functions with an addition formula (F(x+y)=P(F(x),F(y)) P a polynomial) are elliptic functions
		Formal definition of the sine function (via integrals) and derivation of some of its properties.
	www.math.niu.edu /~rusin/known-math/index/33-XX.html (611 words)

	[No title]
		The Euler Gamma function is defined by $\Gamma(z)=\int_{0}^{\infty} t^{z-1}e^{-t} dt$.
		The associated Legendre function $P_{\nu}^{\mu}(z)$ is defined by $P_{\nu}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)} (\frac{z+1}{z-1})^{\frac{1}{2}\mu}F(-\nu, \nu+1; 1-\mu; \frac{1-z}{2})$ where $F(a, b; c;z)$ is a hypergeometric function and $1-z\lt2$.
		The hypergeometric function $F(a, b; c; z)$ is the solution of the hypergeometric differential equation $z(1-z)w^{\prime\prime}+[c-(a+b+1)z]w^{\prime}-abw=0$.
	www.csd.uwo.ca /faculty/watt/home/research/openmath/spfun1.ocd (3083 words)

	/usr/share/local/matlab42/toolbox/stats Directory Documentation
		function with parameters A and B at the values in X. The size of P is the common size of the input arguments.
		function with parameters A and B at the values in X. The size of Y is the common size of the input arguments.
		function with parameters N and P at the values in X. The size of Y is the common size of the input arguments.
	bass.gmu.edu /matlab/toolbox/tstats.html (6046 words)

	Amazon.com: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables: Books: Milton ... (Site not responding. Last check: 2007-11-03)
		First, because algorithms for computing numerical values of mathematically functions have improved dramatically over the 37 years since this work was published, you will not find suitable algorithms for computing values of the various functions discussed.
		Second, and for much the same reason, you should not assume that all the numerical values given in all the tables are completely accurate; in 1964 calculations of some of these values with then-known algorithms pushed the state of the art to the limit.
		For example, in Table 7.3, "Complementary Error Function", two of the values attributed to a 1951 table by O. Emersleben are slightly incorrect in the last digit tabulated.
	www.amazon.com /Handbook-Mathematical-Functions-Formulas-Tables/dp/0486612724 (1427 words)

	Mathematics and Statistics--Best Sites
		Mathematical Atlas: introduction to a variety of concepts; uses Mathematical Reviews' Classifications.
		Canadian Society for History and Philosophy of Mathematics: the society's bulletin and the History and Pedagogy of Mathematics Newsletter are available in WordPerfect format.
		New York Journal of Mathematics: articles are available in pdf, postscript, dvi and hyperdvi.
	www.southalabama.edu /univlib/picks/math.htm (618 words)

	Mathematics of Computation
		Abstract: The Riemann theta function is a complex-valued function of
		First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision.
		Handbook of mathematical functions, with formulas, graphs and mathematical tables, National Bureau of Standards, Washington, D.C., 1966, Edited by M. Abramowitz and I. Stengun.
	www.ams.org /mcom/2004-73-247/S0025-5718-03-01609-0/home.html (440 words)

	Toward a Revised NBS Handbook of Mathematical Functions
		A modernized and updated revision of Abramowitz and Stegun, Handbook of Mathematical Functions, first published in 1964 by the National Bureau of Standards, is being planned for publication on the World Wide Web.
		This paper was presented at the SIAM Annual Meeting at Stanford University on July 14, 1997, in the Minisymposium on Handbooks for Special Functions and the World Wide Web.
		Comments on any aspect of a revised Handbook of Mathematical Functions will be gratefully received.
	dlmf.nist.gov /about/publications/nistir6072 (197 words)

	DIGITAL LIBRARY OF MATHEMATICAL FUNCTIONS
		These functions (often known as special functions) are used extensively in mathematical analysis in many fields, such as physics and chemistry, and they are essential tools in modern computational modeling of phenomena in the physical sciences and engineering.
		An applications component will be included for electromagnetism and quantum physics to provide access to mathematical information in these fields, to coordinate and standardize definitions and notation, and to serve as a prototype applications component for other fields.
		The core mathematical material for the DLMF is being developed from a thorough review of the relevant literature of the past 40 years and is being enriched by current computational methods and typical scientific applications.
	www.itl.nist.gov /lab/pub/newsaug00.htm (1661 words)

	Dr. Dobb's \| The C Mathematical Function Handbook \| April 15, 2003 (Site not responding. Last check: 2007-11-03)
		A note on the cover claims that the book, C Mathematical Function Handbook, by Louis Baker, is a collection of programming tools for scientists and engineers.
		That is, it generates a list of functions that a given function calls and a list of functions that call a given function.
		I wish he would have broken with the past and conformed to ANSI Standard C. Another complaint about the code is that Baker does not take advantage of the ability to pass structures to functions and, similarly, return a structure from a function.
	www.ddj.com /dept/cpp/184402593 (1146 words)

	Parabolic cylinder functions - Wikipedia, the free encyclopedia
		In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation
		For half-integer values of a, these can be re-expressed in terms of Hermite polynomials; alternately, they can also be expressed in terms of Bessel functions.
		Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1972, Dover: New York.
	en.wikipedia.org /wiki/Parabolic_cylinder_functions (148 words)

	Math web sites
		History of Mathematics bibliography of books and internet sources on the history of mathematics.
		MacTutor History of Mathematics articles on math history and biographies with female mathematicians.
		Digital library of Mathematical Functions National Institute of Standards and Technology is developing a replacement for the
	library.cord.edu /borchert/math/sites.htm (209 words)

	NIST Digital Library of Mathematical Functions
		Starting point of the project is to review and update the Handbook of Mathematical Functions (with Formulas, Graphs, and Mathematical Tables) edited by M. Abramowitz and I.A. Stegun.
		This handbook was first published by the National Bureau of Standards (NBS) in 1964 and last corrected in 1972.
		It should turn out to be a major new mathematical reference source on the World Wide Web for special functions and their applications.
	aw.twi.tudelft.nl /~koekoek/dlmf.html (306 words)

	Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
		It contains a very complete table of analytical integrals, differential equations, and numerical series; and includes tables of trigonometric and hyperbolic functions, tables for numerical integration, rules for differentiation and integration, and techniques for point interpolation and function approximation.
		Additionally, the book devotes an entire section to mathematical and physical constants as fractions and powers of Pi, e, and prime numbers; and discusses statistics by presenting combinatorial analysis and probability functions.
		Elementary Transcendental Functions Logarithmic, Exponential, Circular and Hyperbolic Functions
	www.knovel.com /knovel2/Toc.jsp?BookID=528 (367 words)

	Functions - Functions (I) (Site not responding. Last check: 2007-11-03)
		Integral transforms and special functions belong to the basic subjects of mathematical analysis, the theory of differential and integral equations,
		Trigonometric functions are often omitted in basic calculus courses for students However, the sine and cosine functions are extremely useful in modeling
		Using functions we can structure our programs in a more modular way, accessing all the potential that And here is another example about functions:
	pilotorg.com /?q=functions (223 words)

	MATHFUNC (Site not responding. Last check: 2007-11-03)
		It turns out that the Ai(x) function is generated by going from a point 'a' at infinity in the third quadrant to point 'b' at infinity in the second quadrant of the t plane as indicated by the magenta colored curve.
		His paper, "An Essay on the Application of Mathematical Analyisis to Theories of Electricity and Magnetism", brought him to the attention of the scientific establishment and he was invited, and then began, to attend Cambridge University as an undergraduate at the ripe old age of forty.
		The explicit form of Green's function becomes G=-(1/(2Pi))[ ln(r)-ln(r')], where r and r' are the distances from P and P' to a point Q on the x axis.
	aemes.mae.ufl.edu /~uhk/MATHFUNC.htm (15365 words)

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