Laplace transform with respect to lattice parameter

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11: 7.14 Integrals

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Fourier Transform

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Laplace Transforms

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7.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\left(bt\right)dt=\frac{1}{a}{\mathrm{e}}^{a^2/(4b^2)}\mathrm{erfc}\left(\frac{a}{2b}\right),$$ $\mathrm{\Re }a>0$, ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplacetransform

A&S Ref:

7.4.17

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E2

Encodings:

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See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{(a-b)t}\mathrm{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t}}\right)dt=\frac{{\mathrm{e}}^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})},$$ , $\mathrm{\Re }b>0$, $\mathrm{\Re }c\ge 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplacetransform

A&S Ref:

7.4.21

Referenced by:

§3.5(ix), §7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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Laplace Transforms

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12: 2.3 Integrals of a Real Variable

… ►Assume that the Laplace transform …Then … ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►

§2.3(iii) Laplace’s Method

… ►When $x\to +\mathrm{\infty }$ Laplace’s method (§2.3(iii)) applies, but the form of the resulting approximation is discontinuous at $\alpha =0$. …

13: 29.19 Physical Applications

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§29.19(i) Lamé Functions

►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. … ►Shail (1978) treats applications to solutions of elliptic crack and punch problems. …

14: 3.11 Approximation Techniques

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Laplace Transform Inversion

►Numerical inversion of the Laplace transform (§1.14(iii)) …requires $f={ℒ}^{-1}F$ to be obtained from numerical values of $F$. A general procedure is to approximate $F$ by a rational function $R$ (vanishing at infinity) and then approximate $f$ by $r={ℒ}^{-1}R$. … ►

The Fast Fourier Transform

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15: 11.7 Integrals and Sums

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11.7.10 $${\int }_0^{\mathrm{\infty }}{t}^{-\nu -1}{𝐇}_{\nu }\left(t\right)dt=\frac{\pi }{2^{\nu +1}\mathrm{\Gamma }\left(\nu +1\right)},$$ $\mathrm{\Re }\nu >-\frac{3}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: real or complex order

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/11.7.E10

Encodings:

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See also:

Annotations for §11.7(ii), §11.7 and Ch.11

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§11.7(iii) Laplace Transforms

►The following Laplace transforms of ${𝐇}_{\nu }\left(t\right)$ require $\mathrm{\Re }a>0$ for convergence, while those of ${𝐋}_{\nu }\left(t\right)$ require $\mathrm{\Re }a>1$. … ►

§11.7(iv) Integrals with Respect to Order

►For integrals of ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ with respect to the order $\nu$, see Apelblat (1989). …

16: 13.23 Integrals

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§13.23(i) Laplace and Mellin Transforms

… ►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). ►

§13.23(ii) Fourier Transforms

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§13.23(iii) Hankel Transforms

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17: 9.10 Integrals

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§9.10(v) Laplace Transforms

… ►For Laplace transforms of products of Airy functions see Shawagfeh (1992). ►

§9.10(vi) Mellin Transform

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§9.10(vii) Stieltjes Transforms

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§9.10(ix) Compendia

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18: 31.2 Differential Equations

… ►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). … ►

$F$-Homotopic Transformations

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Homographic Transformations

… ►If $\tilde{z}=\tilde{z}(z)$ is one of the $3!=6$ homographies that map $\mathrm{\infty }$ to $\mathrm{\infty }$, then $w(z)=\tilde{w}(\tilde{z})$ satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and appropriately transformed parameters. … ►

Composite Transformations

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19: Bibliography W

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The analyticity of Jacobian functions with respect to the parameter $k$ . Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§22.17(ii), §22.2.

Encodings:

BibTeX

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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External Links:

ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§22.4(i).

Encodings:

BibTeX

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Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.

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External Links:

ISSN 1687-1847, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§16.6, §16.6.

Encodings:

BibTeX

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.

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External Links:

FullText, MathReview (J. D. Tamarkin), ZentralBlatt

Referenced by:

§1.14(vi), §1.15(viii).

Encodings:

BibTeX

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20: 16.5 Integral Representations and Integrals

… ►Suppose first that $L$ is a contour that starts at infinity on a line parallel to the positive real axis, encircles the nonnegative integers in the negative sense, and ends at infinity on another line parallel to the positive real axis. … ►Secondly, suppose that $L$ is a contour from $-\mathrm{i}\mathrm{\infty }$ to $\mathrm{i}\mathrm{\infty }$. … ►Lastly, the restrictions on the parameters can be eased by replacing the integration paths with loop contours; see Luke (1969a, §3.6). ►Laplace transforms and inverse Laplace transforms of generalized hypergeometric functions are given in Prudnikov et al. (1992a, §3.38) and Prudnikov et al. (1992b, §3.36). …