error and related functions
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11: 7.21 Physical Applications
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►The error functions, Fresnel integrals, and related functions occur in a variety of physical applications.
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12: 7.19 Voigt Functions
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7.19.2
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13: 11.10 Anger–Weber Functions
§11.10 Anger–Weber Functions
… ►§11.10(vi) Relations to Other Functions
… ► … ►§11.10(viii) Expansions in Series of Products of Bessel Functions
… ►§11.10(ix) Recurrence Relations and Derivatives
…14: 15.4 Special Cases
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§15.4(i) Elementary Functions
… ► ►§15.4(ii) Argument Unity
… ►Chu–Vandermonde Identity
… ►§15.4(iii) Other Arguments
…15: Software Index
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support.
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►In the list below we identify four main sources of software for computing special functions.
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Commercial Software.
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►The following are web-based software repositories with significant holdings in the area of special functions.
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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9 Airy and Related Functions | |||||||||||||||||||||||||
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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.
16: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►§8.17(ii) Hypergeometric Representations
… ►§8.17(iii) Integral Representation
… ►§8.17(iv) Recurrence Relations
… ►§8.17(vi) Sums
…17: 8.27 Approximations
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DiDonato (1978) gives a simple approximation for the function (which is related to the incomplete gamma function by a change of variables) for real and large positive . This takes the form , approximately, where and is shown to produce an absolute error as .
18: 7.10 Derivatives
§7.10 Derivatives
…19: Bibliography S
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Chebyshev expansions for the error and related functions.
Math. Comp. 32 (144), pp. 1232–1240.
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Automatic computing methods for special functions. IV. Complex error function, Fresnel integrals, and other related functions.
J. Res. Nat. Bur. Standards 86 (6), pp. 661–686.
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20: Bibliography Z
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On the Computation of Zeros of Bessel and Bessel-related Functions.
In Proceedings of the Sixth International Colloquium on
Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.),
Utrecht, pp. 409–416.
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The Dilogarithm Function in Geometry and Number Theory.
In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.),
Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.
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Numerical analysis of Struve functions with applications to other special functions.
Ann. Numer. Math. 2 (1-4), pp. 199–208.
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Error bounds for asymptotic solutions of second-order linear difference equations.
J. Comput. Appl. Math. 71 (2), pp. 191–212.
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The summation of series of hyperbolic functions.
SIAM J. Math. Anal. 10 (1), pp. 192–206.
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