integral%20identities
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1: Bibliography O
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Complete elliptic integrals resulting from infinite integrals of Bessel functions.
J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.
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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II.
J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.
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Uniform Airy-type expansions of integrals.
SIAM J. Math. Anal. 25 (2), pp. 304–321.
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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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A class of integral identities with matrix argument.
Duke Math. J. 26 (2), pp. 207–213.
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2: 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 …3: Bibliography K
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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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Cyclic identities for Jacobi elliptic and related functions.
J. Math. Phys. 44 (4), pp. 1822–1841.
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Cyclic identities involving Jacobi elliptic functions.
J. Math. Phys. 43 (7), pp. 3798–3806.
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Dilogarithm identities.
Progr. Theoret. Phys. Suppl. (118), pp. 61–142.
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4: 9.18 Tables
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Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
Yakovleva (1969) tabulates Fock’s functions , , , for . Precision is 7S.
Sherry (1959) tabulates , , , , ; 20S.
§9.18(v) Integrals
… ►National Bureau of Standards (1958) tabulates and for and ; for . Precision is 8D.
5: 18.40 Methods of Computation
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►Given the power moments, , , can these be used to find a unique , a non-decreasing, real, function of , in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant.
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18.40.4
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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►Equation (18.40.7) provides step-histogram approximations to , as shown in Figure 18.40.1 for and , shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein.
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►The bottom and top of the steps at the are lower and upper bounds to as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43).
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6: 20.11 Generalizations and Analogs
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►This is the discrete analog of the Poisson identity (§1.8(iv)).
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►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).
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►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).
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►Similar identities can be constructed for , , and .
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7: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Rogers-Ramanujan identities in the hard hexagon model.
J. Statist. Phys. 26 (3), pp. 427–452.
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Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics.
In Proceedings of the International Congress of Mathematicians,
Vol. III (Berlin, 1998),
pp. 163–172.
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Phase-space projection identities for diffraction catastrophes.
J. Phys. A 13 (1), pp. 149–160.
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A cubic counterpart of Jacobi’s identity and the AGM.
Trans. Amer. Math. Soc. 323 (2), pp. 691–701.
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8: 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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Some identities involving the Riemann zeta function. II.
Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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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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9: Bibliography G
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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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A table of integrals of the exponential integral.
J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.
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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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A new identity for some six- symbols.
J. Math. Phys. 32 (6), pp. 1430–1432.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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