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21: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…22: 25.12 Polylogarithms
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►The right-hand side is called Clausen’s integral.
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Integral Representation
… ►§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals
►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms …23: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►Addendum: For a companion equation see (8.17.24). … ►§8.17(ii) Hypergeometric Representations
… ►§8.17(iii) Integral Representation
… ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. …24: 9.18 Tables
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Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
Sherry (1959) tabulates , , , , ; 20S.
Zhang and Jin (1996, p. 339) tabulates , , , , , , , , ; 8D.
§9.18(v) Integrals
…25: 6.20 Approximations
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Cody and Thacher (1968) provides minimax rational approximations for , with accuracies up to 20S.
Cody and Thacher (1969) provides minimax rational approximations for , with accuracies up to 20S.
MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions and , with accuracies up to 20S.
Luke (1969b, pp. 41–42) gives Chebyshev expansions of , , and for , . The coefficients are given in terms of series of Bessel functions.
26: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
… ►The Riemann zeta function is a special case: … ►§25.11(vii) Integral Representations
… ►§25.11(viii) Further Integral Representations
… ►§25.11(ix) Integrals
…27: Peter L. Walker
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►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004.
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28: 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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Integrals of Airy Functions.
National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Generalization of Binet’s Gamma function formulas.
Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
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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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29: 36.5 Stokes Sets
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