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1: 19.36 Methods of Computation
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►The incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6).
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►If the iteration of (19.36.6) and (19.36.12) is stopped when ( and being approximated by and , and the infinite series being truncated), then the relative error in and is less than if we neglect terms of order .
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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►For series expansions of Legendre’s integrals see §19.5.
Faster convergence of power series for and can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12).
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2: 23.9 Laurent and Other Power Series
3: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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On the power function of the likelihood ratio test for MANOVA.
J. Multivariate Anal. 82 (2), pp. 416–421.
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A short table of the functions , from to
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Phil. Mag. Series 7 20, pp. 343–347.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Formal Power Series and Algebraic Combinatorics.
DIMACS Series in Discrete Mathematics and Theoretical Computer
Science, Vol. 24, American Mathematical Society, Providence, RI.
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4: 20.11 Generalizations and Analogs
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§20.11(ii) Ramanujan’s Theta Function and -Series
… ►In 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). … ►As in §20.11(ii), the modulus of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in -series via (20.9.1). … ►For applications to rapidly convergent expansions for see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). …5: Bibliography L
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New method to obtain small parameter power series expansions of Mathieu radial and angular functions.
Math. Comp. 78 (265), pp. 255–274.
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Moment integrals of powers of Airy functions.
Z. Angew. Math. Phys. 44 (5), pp. 891–908.
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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Integral and series representations of the Dirac delta function.
Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
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6: 2.11 Remainder Terms; Stokes Phenomenon
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►If we permit the use of nonelementary functions as approximants, then even more powerful re-expansions become available.
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►The first of these two references also provides an introduction to the powerful Borel transform theory.
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►The process just used is equivalent to re-expanding the remainder term of the original asymptotic series (2.11.24) in powers of and truncating the new series optimally.
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►The following example, based on Weniger (1996), illustrates their power.
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►For example, using double precision is found to agree with (2.11.31) to 13D.
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7: Bibliography C
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Power series for inverse Jacobian elliptic functions.
Math. Comp. 77 (263), pp. 1615–1621.
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Introduction to the Theory of Fourier’s Series and Integrals.
3rd edition, Macmillan, London.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems.
SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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8: Bibliography K
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The computation of the sums of negative even powers of roots of Bessel functions.
Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively -binomial sums and basic hypergeometric series.
Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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9: 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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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Multiplicative processes and power laws.
Phys. Rev. E 57 (4), pp. 4811–4813.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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An Introduction to Basic Fourier Series.
Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
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10: Bibliography G
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Contiguous relations and summation and transformation formulae for basic hypergeometric series.
J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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A monotonicity property of the power function of multivariate tests.
Indag. Math. (N.S.) 11 (2), pp. 209–218.
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