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1: 4.22 Infinite Products and Partial Fractions
§4.22 Infinite Products and Partial Fractions
2: 4.36 Infinite Products and Partial Fractions
§4.36 Infinite Products and Partial Fractions
3: Bibliography R
  • J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • R. Reynolds and A. Stauffer (2021) Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).
  • M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
  • M. Rothman (1954b) The problem of an infinite plate under an inclined loading, with tables of the integrals of Ai ( ± x ) and Bi ( ± x ) . Quart. J. Mech. Appl. Math. 7 (1), pp. 1–7.
  • 4: 20 Theta Functions
    Chapter 20 Theta Functions
    5: 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. D. Kerr (1978) An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • 6: 3.8 Nonlinear Equations
    For the computation of zeros of Bessel functions, Coulomb functions, and conical functions as eigenvalues of finite parts of infinite tridiagonal matrices, see Grad and Zakrajšek (1973), Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil et al. (2007a, pp. 205–213). …
    3.8.15 p ( x ) = ( x 1 ) ( x 2 ) ( x 20 )
    Consider x = 20 and j = 19 . We have p ( 20 ) = 19 ! and a 19 = 1 + 2 + + 20 = 210 . …
    3.8.16 d x d a 19 = 20 19 19 ! = ( 4.30 ) × 10 7 .
    7: 1.9 Calculus of a Complex Variable
    §1.9(v) Infinite Sequences and Series
    This sequence converges pointwise to a function f ( z ) if …
    §1.9(vii) Inversion of Limits
    Dominated Convergence Theorem
    8: 25.12 Polylogarithms
    25.12.8 n = 1 cos ( n θ ) n 2 = π 2 6 π θ 2 + θ 2 4 .
    25.12.9 n = 1 sin ( n θ ) n 2 = 0 θ ln ( 2 sin ( 1 2 x ) ) d x .
    See accompanying text
    Figure 25.12.1: Dilogarithm function Li 2 ( x ) , 20 x < 1 . Magnify
    See accompanying text
    Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . … Magnify 3D Help
    25.12.10 Li s ( z ) = n = 1 z n n s .
    9: 1.10 Functions of a Complex Variable
    §1.10(ix) Infinite Products
    The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. …
    §1.10(x) Infinite Partial Fractions
    Mittag-Leffler’s Expansion
    10: 1.5 Calculus of Two or More Variables
    Infinite Integrals
    Suppose that a , b , c are finite, d is finite or + , and f ( x , y ) , f / x are continuous on the partly-closed rectangle or infinite strip [ a , b ] × [ c , d ) . …
    Infinite Double Integrals
    Infinite double integrals occur when f ( x , y ) becomes infinite at points in D or when D is unbounded. … Finite and infinite integrals can be defined in a similar way. …