rational functions
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11: 6.7 Integral Representations
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βΊMany integrals with exponentials and rational functions, for example, integrals of the type , where is an arbitrary rational function, can be represented in finite form in terms of the function
and elementary functions; see Lebedev (1965, p. 42).
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12: Bibliography
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A -beta integral on the unit circle and some biorthogonal rational functions.
Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
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Rational function approximations for Fermi-Dirac integrals.
The Astrophysical Journal Supplement Series 84, pp. 101–108.
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13: 20.11 Generalizations and Analogs
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βΊIn the case identities for theta functions become identities in the complex variable , with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
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14: 17.3 -Elementary and -Special Functions
15: Bibliography W
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Rational approximations for the modified Bessel function of the second kind.
Comput. Phys. Comm. 59 (3), pp. 471–493.
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A rational approximant for the digamma function.
Numer. Algorithms 33 (1-4), pp. 499–507.
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Rational function certification of multisum/integral/“” identities.
Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.
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Rational Chebyshev approximations for the Bessel functions
, , ,
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Math. Comp. 39 (160), pp. 617–623.
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16: 7.24 Approximations
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Hastings (1955) gives several minimax polynomial and rational approximations for , and the auxiliary functions and .
Cody (1969) provides minimax rational approximations for and . The maximum relative precision is about 20S.
Cody et al. (1970) gives minimax rational approximations to Dawson’s integral (maximum relative precision 20S–22S).
17: 7.7 Integral Representations
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βΊIntegrals of the type , where is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions.
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18: 15.5 Derivatives and Contiguous Functions
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βΊBy repeated applications of (15.5.11)–(15.5.18) any function
, in which are integers, can be expressed as a linear combination of and any one of its contiguous functions, with coefficients that are rational functions of , and .
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