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21: 13.23 Integrals
§13.23(i) Laplace and Mellin Transforms
13.23.1 0 e - z t t ν - 1 M κ , μ ( t ) d t = Γ ( μ + ν + 1 2 ) ( z + 1 2 ) μ + ν + 1 2 F 1 2 ( 1 2 + μ - κ , 1 2 + μ + ν 1 + 2 μ ; 1 z + 1 2 ) , μ + ν + 1 2 > 0 , z > 1 2 .
13.23.2 0 e - z t t μ - 1 2 M κ , μ ( t ) d t = Γ ( 2 μ + 1 ) ( z + 1 2 ) - κ - μ - 1 2 ( z - 1 2 ) κ - μ - 1 2 , μ > - 1 2 , z > 1 2 ,
For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), and Prudnikov et al. (1992a, §§3.34, 3.35). Inverse Laplace transforms are given in Oberhettinger and Badii (1973, §2.16) and Prudnikov et al. (1992b, §§3.33, 3.34). …
22: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
The Laplace transform of ϕ ( ρ , β ; z ) can be expressed in terms of the Mittag-Leffler function: …
23: 18.17 Integrals
§18.17(vi) Laplace Transforms
Jacobi
Laguerre
Hermite
18.17.40 0 e - a x L n ( α ) ( b x ) x z - 1 d x = Γ ( z + n ) n ! ( a - b ) n a - n - z F 1 2 ( - n , 1 + α - z 1 - n - z ; a a - b ) , a > 0 , z > 0 .
24: Bibliography P
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992a) Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.
  • 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.
  • 25: 7.7 Integral Representations
    7.7.4 0 e - a t t + z 2 d t = π a e a z 2 erfc ( a z ) , a > 0 , z > 0 .
    7.7.15 0 e - a t cos ( t 2 ) d t = π 2 f ( a 2 π ) , a > 0 ,
    7.7.16 0 e - a t sin ( t 2 ) d t = π 2 g ( a 2 π ) , a > 0 .
    26: 3.5 Quadrature
    Example. Laplace Transform Inversion
    3.5.38 G ( p ) = 0 e - p t g ( t ) d t ,
    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)): …
    27: Bibliography H
  • P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
  • R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.
  • 28: 35.8 Generalized Hypergeometric Functions of Matrix Argument
    Laplace Transform
    29: 8.19 Generalized Exponential Integral
    8.19.2 E p ( z ) = z p - 1 z e - t t p d t .
    8.19.4 E p ( z ) = z p - 1 e - z Γ ( p ) 0 t p - 1 e - z t 1 + t d t , | ph z | < 1 2 π , p > 0 .
    30: Errata
  • Section 1.14

    There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

    Transform New Abbreviated Old
    Notation Notation Notation
    Fourier ( f ) ( x ) f ( x )
    Fourier Cosine c ( f ) ( x ) c f ( x )
    Fourier Sine s ( f ) ( x ) s f ( x )
    Laplace ( f ) ( s ) f ( s ) ( f ( t ) ; s )
    Mellin ( f ) ( s ) f ( s ) ( f ; s )
    Hilbert ( f ) ( s ) f ( s ) ( f ; s )
    Stieltjes 𝒮 ( f ) ( s ) 𝒮 f ( s ) 𝒮 ( f ; s )

    Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.