inverse Laplace transforms

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11—20 of 23 matching pages

11: Bibliography P

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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.

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External Links:: ISBN 2-88124-838-1, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §13.23(iv), §14.17(v), §15.14, §15.14, §16.15, §16.20, §16.5, §18.17(ix), §19.13(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.6(iii), §9.10(ix).
Encodings:: BibTeX

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… ►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

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Example. Laplace Transform Inversion

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Table 3.5.19: Laplace transform inversion.

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$t$	$J_{0}(t)$	$g(t)$
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► … ►In fact from (7.14.4) and the inversion formula for the Laplace transform (§1.14(iii)) we have … ►A special case is the rule for Hilbert transforms (§1.14(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

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Inversion

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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

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H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:: FullText, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

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18: Bibliography K

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V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:: Translated from the Russian by George Yankovsky. Reprint of the 1977 translation
External Links:: MathReview, ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

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19: 10.43 Integrals

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10.43.22

\int_{0}^{\infty}t^{\mu-1}e^{-at}K_{\nu}\left(t\right)\,\mathrm{d}t=\begin{% cases}\left(\frac{1}{2}\pi\right)^{\frac{1}{2}}\Gamma\left(\mu-\nu\right)% \Gamma\left(\mu+\nu\right)(1-a^{2})^{-\frac{1}{2}\mu+\frac{1}{4}}\mathsf{P}^{-% \mu+\frac{1}{2}}_{\nu-\frac{1}{2}}\left(a\right),&-1<a<1,\\ \left(\frac{1}{2}\pi\right)^{\frac{1}{2}}\Gamma\left(\mu-\nu\right)\Gamma\left% (\mu+\nu\right)(a^{2}-1)^{-\frac{1}{2}\mu+\frac{1}{4}}P^{-\mu+\frac{1}{2}}_{% \nu-\frac{1}{2}}\left(a\right),&\Re a\geq 0,a\neq 1.\end{cases}

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§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

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§6.14(i) Laplace Transforms

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6.14.3

\int_{0}^{\infty}e^{-at}\operatorname{si}\left(t\right)\,\mathrm{d}t=-\frac{1}% {a}\operatorname{arctan}a,

\Re a>0

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $\operatorname{si}\left(\NVar{z}\right)$ : sine integral
A&S Ref:: 5.2.29
Permalink:: http://dlmf.nist.gov/6.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.14(i), §6.14 and Ch.6

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