Laplace transform with respect to lattice parameter

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(iii) Laplace Transform

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Inversion

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Translation

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Derivatives

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2: 19.13 Integrals of Elliptic Integrals

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§19.13(i) Integration with Respect to the Modulus

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§19.13(ii) Integration with Respect to the Amplitude

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for $F(\varphi ,k)$ and $E(\varphi ,k)$, together with special cases. ►

§19.13(iii) Laplace Transforms

►For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.

3: 35.2 Laplace Transform

§35.2 Laplace Transform

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Definition

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Inversion Formula

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Convolution Theorem

►If $g_j$ is the Laplace transform of $f_j$, $j=1,2$, then $g_1{g}_2$ is the Laplace transform of the convolution $f_1\ast f_2$, where …

4: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

… ►Then …

5: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). Inverse Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §5.19), Oberhettinger and Badii (1973, §2.18), and Prudnikov et al. (1992b, §3.35). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

6: 14.31 Other Applications

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§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

7: 16.20 Integrals and Series

… ►Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer $G$-function are given in Prudnikov et al. (1992a, §3.40) and Prudnikov et al. (1992b, §3.38). …

8: 2.5 Mellin Transform Methods

… ►To apply the Mellin transform method outlined in §2.5(i), we require the transforms $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ to have a common strip of analyticity. … ►Since $ℳh_1\left(z\right)$ is analytic for $\mathrm{\Re }z>-c$ by Table 2.5.1, the analytically-continued $ℳh_2\left(z\right)$ allows us to extend the Mellin transform of $h$ via … ►

§2.5(iii) Laplace Transforms with Small Parameters

►Let $h(t)$ satisfy (2.5.18) and (2.5.20) with $c>-1$, and consider the Laplace transform … ►For examples in which the integral defining the Mellin transform $ℳh\left(z\right)$ does not exist for any value of $z$, see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).

9: 6.14 Integrals

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§6.14(i) Laplace Transforms

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6.14.5 $${\int }_0^{\mathrm{\infty }}\cos t\mathrm{Ci}\left(t\right)dt={\int }_0^{\mathrm{\infty }}\sin t\mathrm{si}\left(t\right)dt=-\frac{1}{4}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral, $\mathrm{si}\left(z\right)$: sine integral and $\sin z$: sine function

A&S Ref:

5.2.30

Permalink:

http://dlmf.nist.gov/6.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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6.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{Ci}}^2\left(t\right)dt={\int }_0^{\mathrm{\infty }}{\mathrm{si}}^2\left(t\right)dt=\frac{1}{2}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.31

Permalink:

http://dlmf.nist.gov/6.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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10: 2.4 Contour Integrals

… ►Then … ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). ►

§2.4(ii) Inverse Laplace Transforms

… ►Then the Laplace transform … ►For examples see Olver (1997b, pp. 315–320). …