Integration in Finite Terms with Special Functions: the Error Function

Cherry, Guy W.

doi:10.1016/s0747-7171(85)80037-7

Cited by 22 publications

(8 citation statements)

References 6 publications

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“…In recent years, the algorithmic evaluation of indefinite integrals involving elementary functions and some non-elementary extensions has been sfudied and well established [2,4,5,16,19]. However, in many applications there arise definite integrals which can be expressed in closed form (often in terms of special functions) for which the corresponding indefinite integral cannot be expressed in closed form.…”

Section: Introductionmentioning

confidence: 99%

Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions

Geddes

Glasser

Moore³

et al. 1990

AAECC

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Abstract. Herein, it is shown that by exploiting integral definitions of well known special functions, through generalizations and differentiations, broad classes of definite integrals can be solved in closed form or in terms of special functions. This is especially useful when there is no closed form solution to the indefinite form of the integral. In this paper, three such classes of definite integrals are presented. Two of these classes incorporate and supercede all of K61big's integration formulae [11], including his formulation for the computation of Cauchy principal values. Also presented are the mathematical derivations that support the implementation of a third class which exploits the incomplete Gamma function. The resulting programs, based on pattern matching, differentiation, and occasionally limits, are very efficient.

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Section: Introductionmentioning

confidence: 99%

Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions

Geddes

Glasser

Moore³

et al. 1990

AAECC

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“…Generally,w em ay allows pecial functions as extensions from K to L that satisfy certain differential equations. In case of y =e rf (g)t his is y′ = g′ exp(−g 2 ), ignoring the scaling factor.D ecision procedures based on Liouville type theorems and the Risch approach have been devised in this setting, but here we can only refer to Singer et al (1985) [159], Cherry (1985 [36], 1986) [35], and Knowles (1986) [92].…”

Section: Then F N Is Called An Elementary Liouvillian Extension Of C(z)mentioning

confidence: 99%

Computer Algebra Algorithms

Kaltofen¹

1987

Annu. Rev. Comput. Sci.

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“…Cherry [4] was able to produce a decision procedure to integrate transcendental elementary functions in terms of elementary functions and logarithmic integrals. He , [3], also produced a decision procedure for integrating a class of transcendental elementary functions in terms of elementary functions and error functions. Knowles [5] generalized Cherry's results allowing error functions and logarithmic integrals to occur in both the integrand and the integral along with elementary functions.…”

Section: Introductionmentioning

confidence: 99%

A Note on Symbolic Integration with Polylogarithms

Baddoura¹

2010

Mediterr. J. Math.

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We generalize partially Liouville's theorem on integration in finite terms to allow polylogarithms of any order to occur in the integral in addition to elementary functions. The result is a partial generalization of a theorem proved by the author for the dilogarithm. It is also a partial proof of a conjecture postulated by the author in 1994 [1]. The basic conclusion is that an associated function to the nth polylogarithm appears linearly with logarithms appearing possibly in a polynomial way with non-constant coefficients. Mathematics Subject Classification (2010). 99Z99; 12H05; 33F10.

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Integration in Finite Terms with Special Functions: the Error Function

Cited by 22 publications

References 6 publications

Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions

Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions

Computer Algebra Algorithms

A Note on Symbolic Integration with Polylogarithms

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