About the Project

integral%20identities

AdvancedHelp

(0.002 seconds)

1—10 of 13 matching pages

1: Bibliography O
  • S. Okui (1974) Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.
  • S. Okui (1975) Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.
  • A. B. Olde Daalhuis and N. M. Temme (1994) Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • I. Olkin (1959) A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.
  • 2: 25.12 Polylogarithms
    The right-hand side is called Clausen’s integral. …
    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
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.
  • A. Khare and U. Sukhatme (2002) Cyclic identities involving Jacobi elliptic functions. J. Math. Phys. 43 (7), pp. 3798–3806.
  • A. N. Kirillov (1995) Dilogarithm identities. Progr. Theoret. Phys. Suppl. (118), pp. 61–142.
  • 4: 9.18 Tables
  • Zhang and Jin (1996, p. 337) tabulates Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) for x = 0 ( 1 ) 20 to 8S and for x = 20 ( 1 ) 0 to 9D.

  • Yakovleva (1969) tabulates Fock’s functions U ( x ) π Bi ( x ) , U ( x ) = π Bi ( x ) , V ( x ) π Ai ( x ) , V ( x ) = π Ai ( x ) for x = 9 ( .001 ) 9 . Precision is 7S.

  • Sherry (1959) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; 20S.

  • §9.18(v) Integrals
  • National Bureau of Standards (1958) tabulates A 0 ( x ) π Hi ( x ) and A 0 ( x ) π Hi ( x ) for x = 0 ( .01 ) 1 ( .02 ) 5 ( .05 ) 11 and 1 / x = 0.01 ( .01 ) 0.1 ; 0 x A 0 ( t ) d t for x = 0.5 , 1 ( 1 ) 11 . Precision is 8D.

  • 5: 18.40 Methods of Computation
    Given the power moments, μ n = a b x n d μ ( x ) , n = 0 , 1 , 2 , , can these be used to find a unique μ ( x ) , a non-decreasing, real, function of x , in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. …
    18.40.4 lim N F N ( z ) = F ( z ) 1 μ 0 a b w ( x ) d x z x , z \ [ a , b ] ,
    Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . … Equation (18.40.7) provides step-histogram approximations to a x d μ ( x ) , as shown in Figure 18.40.1 for N = 12 and 120 , shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. … The bottom and top of the steps at the x i are lower and upper bounds to a x i d μ ( x ) as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43). …
    6: 20.11 Generalizations and Analogs
    This is the discrete analog of the Poisson identity1.8(iv)). … In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , 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 k of elliptic integrals19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in q -series via (20.9.1). … Similar identities can be constructed for F 1 2 ( 1 3 , 2 3 ; 1 ; k 2 ) , F 1 2 ( 1 4 , 3 4 ; 1 ; k 2 ) , and F 1 2 ( 1 6 , 5 6 ; 1 ; k 2 ) . …
    7: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.
  • A. Berkovich and B. M. McCoy (1998) 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.
  • M. V. Berry and F. J. Wright (1980) Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.
  • J. M. Borwein and P. B. Borwein (1991) A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.
  • 8: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • R. Sitaramachandrarao and B. Davis (1986) Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • 9: Bibliography G
  • W. Gautschi (1994) 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.
  • M. Geller and E. W. Ng (1969) A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • J. N. Ginocchio (1991) A new identity for some six- j symbols. J. Math. Phys. 32 (6), pp. 1430–1432.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • 10: 25.5 Integral Representations
    §25.5 Integral Representations
    25.5.5 ζ ( s ) = s 0 x x 1 2 x s + 1 d x , 1 < s < 0 .
    25.5.6 ζ ( s ) = 1 2 + 1 s 1 + 1 Γ ( s ) 0 ( 1 e x 1 1 x + 1 2 ) x s 1 e x d x , s > 1 .
    25.5.14 ω ( x ) n = 1 e n 2 π x = 1 2 ( θ 3 ( 0 | i x ) 1 ) .
    §25.5(iii) Contour Integrals