power-series%20expansion
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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: 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). …4: 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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New series expansions for the confluent hypergeometric function
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Appl. Math. Comput. 235, pp. 26–31.
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New series expansions of the Gauss hypergeometric function.
Adv. Comput. Math. 39 (2), pp. 349–365.
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Expansion of the confluent hypergeometric function in series of Bessel functions.
Math. Tables Aids Comput. 13 (68), pp. 261–271.
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5: 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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