error%20functions
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1: 7.24 Approximations
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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).
2: Bibliography N
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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3: Bibliography D
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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4: 7.23 Tables
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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
5: 5.11 Asymptotic Expansions
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►Wrench (1968) gives exact values of up to .
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§5.11(ii) Error Bounds and Exponential Improvement
… ►For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b). … ►For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990). … ►For realistic error bounds in (5.11.14) see Frenzen (1987a, 1992). …6: Bibliography F
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Complex zeros of the error function and of the complementary error function.
Math. Comp. 27 (122), pp. 401–407.
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Error bounds for asymptotic expansions of the ratio of two gamma functions.
SIAM J. Math. Anal. 18 (3), pp. 890–896.
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Error bounds for a uniform asymptotic expansion of the Legendre function
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SIAM J. Math. Anal. 21 (2), pp. 523–535.
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Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument.
SIAM J. Math. Anal. 23 (2), pp. 505–511.
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7: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Chebyshev expansions for the error and related functions.
Math. Comp. 32 (144), pp. 1232–1240.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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On the calculation of the inverse of the error function.
Math. Comp. 22 (101), pp. 144–158.
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8: Bibliography O
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Error analysis of Miller’s recurrence algorithm.
Math. Comp. 18 (85), pp. 65–74.
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Error bounds for asymptotic expansions in turning-point problems.
J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
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Error bounds for stationary phase approximations.
SIAM J. Math. Anal. 5 (1), pp. 19–29.
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Asymptotic approximations and error bounds.
SIAM Rev. 22 (2), pp. 188–203.
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9: Bibliography V
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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions.
CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en
Informatica, Amsterdam.
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An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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Special Functions and the Theory of Group Representations.
American Mathematical Society, Providence, RI.
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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10: 30.9 Asymptotic Approximations and Expansions
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