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inverse Laplace transforms

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11: Bibliography P
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992b) Integrals and Series: Inverse Laplace Transforms, Vol. 5. Gordon and Breach Science Publishers, New York.
  • 12: 14.17 Integrals
    For Laplace transforms and inverse Laplace transforms involving associated Legendre functions, see Erdélyi et al. (1954a, pp. 179–181, 270–272), Oberhettinger and Badii (1973, pp. 113–118, 317–324), Prudnikov et al. (1992a, §§3.22, 3.32, and 3.33), and Prudnikov et al. (1992b, §§3.20, 3.30, and 3.31). …
    13: 3.5 Quadrature
    Example. Laplace Transform Inversion
    Table 3.5.19: Laplace transform inversion.
    t J 0 ( t ) g ( t )
    In fact from (7.14.4) and the inversion formula for the Laplace transform1.14(iii)) we have … A special case is the rule for Hilbert transforms1.14(v)): …
    14: 13.10 Integrals
    Inverse Laplace transforms are given in Oberhettinger and Badii (1973, §2.16) and Prudnikov et al. (1992b, §§3.33, 3.34). …
    15: 1.14 Integral Transforms
    Inversion
    16: 13.23 Integrals
    Inverse Laplace transforms are given in Oberhettinger and Badii (1973, §2.16) and Prudnikov et al. (1992b, §§3.33, 3.34). …
    17: Bibliography S
  • H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.
  • 18: Bibliography K
  • V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.
  • 19: 10.43 Integrals
    10.43.22 0 t μ - 1 e - a t K ν ( t ) d t = { ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( 1 - a 2 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , - 1 < a < 1 , ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( a 2 - 1 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , a 0 , a 1 .
    §10.43(v) Kontorovich–Lebedev Transform
    The Kontorovich–Lebedev transform of a function g ( x ) is defined as … For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996). For collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5). …
    20: 6.14 Integrals
    §6.14(i) Laplace Transforms
    6.14.3 0 e - a t si ( t ) d t = - 1 a arctan a , a > 0 .