About the Project

integral%20representations

AdvancedHelp

(0.003 seconds)

1—10 of 16 matching pages

1: 25.5 Integral Representations
§25.5 Integral Representations
β–Ί
25.5.5 ΢ ⁑ ( s ) = s ⁒ 0 x x 1 2 x s + 1 ⁒ d x , 1 < ⁑ s < 0 .
β–ΊFor similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339). … β–Ί
25.5.19 ΞΆ ⁑ ( m + s ) = ( 1 ) m 1 ⁒ Ξ“ ⁑ ( s ) ⁒ sin ⁑ ( Ο€ ⁒ s ) Ο€ ⁒ Ξ“ ⁑ ( m + s ) ⁒ 0 ψ ( m ) ⁑ ( 1 + x ) ⁒ x s ⁒ d x , m = 1 , 2 , 3 , .
β–Ί
§25.5(iii) Contour Integrals
2: Bibliography M
β–Ί
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • β–Ί
  • O. I. Marichev (1984) On the Representation of Meijer’s G -Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.
  • β–Ί
  • P. Maroni (1995) An integral representation for the Bessel form. J. Comput. Appl. Math. 57 (1-2), pp. 251–260.
  • β–Ί
  • I. MezΕ‘ (2020) An integral representation for the Lambert W function.
  • β–Ί
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • 3: 19.36 Methods of Computation
    §19.36 Methods of Computation
    β–ΊLegendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … β–ΊFor 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). … β–ΊNumerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. … β–Ί
    4: Bibliography V
    β–Ί
  • N. Ja. Vilenkin and A. U. Klimyk (1991) Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.
  • β–Ί
  • N. Ja. Vilenkin and A. U. Klimyk (1993) Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.
  • β–Ί
  • N. Ja. Vilenkin (1968) Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.
  • β–Ί
  • H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.
  • β–Ί
  • H. Volkmer (2021) Fourier series representation of Ferrers function 𝖯 .
  • 5: 25.12 Polylogarithms
    β–ΊThe right-hand side is called Clausen’s integral. … β–Ί
    Integral Representation
    β–Ί
    §25.12(iii) Fermi–Dirac and Bose–Einstein Integrals
    β–ΊThe Fermi–Dirac and Bose–Einstein integrals are defined by … β–ΊIn terms of polylogarithms …
    6: 25.11 Hurwitz Zeta Function
    β–Ί
    §25.11(iii) Representations by the Euler–Maclaurin Formula
    β–Ί
    §25.11(iv) Series Representations
    β–Ί
    §25.11(vii) Integral Representations
    β–Ί
    §25.11(viii) Further Integral Representations
    β–Ί
    §25.11(x) Further Series Representations
    7: Bibliography R
    β–Ί
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • β–Ί
  • W. H. Reid (1995) Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.
  • β–Ί
  • W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.
  • β–Ί
  • W. H. Reid (1997b) Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.
  • β–Ί
  • G. B. Rybicki (1989) Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.
  • 8: 36.5 Stokes Sets
    β–Ί
    §36.5(ii) Cuspoids
    β–Ί
    36.5.7 X = 9 20 + 20 ⁒ u 4 Y 2 20 ⁒ u 2 + 6 ⁒ u 2 ⁒ sign ⁑ ( z ) ,
    β–Ί
    §36.5(iii) Umbilics
    β–Ί
    §36.5(iv) Visualizations
    β–ΊRed and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …
    9: Bibliography C
    β–Ί
  • J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.
  • β–Ί
  • C. Chiccoli, S. Lorenzutta, and G. Maino (1990b) On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.
  • β–Ί
  • M. W. Coffey (2008) On some series representations of the Hurwitz zeta function. J. Comput. Appl. Math. 216 (1), pp. 297–305.
  • β–Ί
  • H. S. Cohl and R. S. Costas-Santos (2020) Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).
  • β–Ί
  • H. S. Cohl (2011) On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.
  • 10: Bibliography S
    β–Ί
  • B. E. Sagan (2001) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.
  • β–Ί
  • R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.
  • β–Ί
  • H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
  • β–Ί
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • β–Ί
  • R. Sips (1949) Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).