Dougall%20expansion
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11: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
… ►§12.11(iii) Asymptotic Expansions for Large Parameter
►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). … ►
12.11.4
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12.11.9
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12: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Applicable Anal. 39 (4), pp. 249–263.
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Methods Appl. Anal. 3 (1), pp. 98–108.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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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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13: 2.11 Remainder Terms; Stokes Phenomenon
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§2.11(iii) Exponentially-Improved Expansions
… ►In this way we arrive at hyperasymptotic expansions. … ► … ►For example, using double precision is found to agree with (2.11.31) to 13D. …14: 16.22 Asymptotic Expansions
§16.22 Asymptotic Expansions
►Asymptotic expansions of for large are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer -functions with large parameters see Fields (1973, 1983).15: 5.11 Asymptotic Expansions
§5.11 Asymptotic Expansions
►§5.11(i) Poincaré-Type Expansions
… ►and … ►Wrench (1968) gives exact values of up to . … ►16: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►§28.8(ii) Sips’ Expansions
… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). ►§28.8(iii) Goldstein’s Expansions
…17: 9.9 Zeros
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§9.9(iv) Asymptotic Expansions
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9.9.6
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9.9.7
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9.9.8
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►For error bounds for the asymptotic expansions of , , , and see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999).
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18: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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Expansions of hypergeometric functions in hypergeometric functions.
Math. Comp. 15 (76), pp. 390–395.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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19: 6.16 Mathematical Applications
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►Compare Figure 6.16.1.
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►It occurs with Fourier-series expansions of all piecewise continuous functions.
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