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31: 29.10 Lamé Functions with Imaginary Periods

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29.10.2 $$z^{\prime }=\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

►transform (29.2.1) into ►

29.10.3 $$\frac{d^{2}w}{{d{z}^{\prime }}^2}+(h^{\prime }-\nu (\nu +1)k_{}^{\prime }{}_{}{}^2{\mathrm{sn}}^2(z^{\prime },k^{\prime }))w=0.$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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32: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(iv) Separation of Variables

►The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi ,\eta ,\varphi )$, admits solutions of the form (30.13.8), where $w_1$ satisfies the differential equation …Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution $z=\pm \mathrm{i}\xi$. …

33: 15.11 Riemann’s Differential Equation

… ►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … ►

15.11.3

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Defines:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation

Symbols:

$z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.4

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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§15.11(ii) Transformation Formulas

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15.11.5 $$t=(\kappa z+\lambda )/(\mu z+\nu ),$$

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Symbols:

$z$: complex variable, $t$: mapping, $\kappa$: constant, $\lambda$: constant, $\mu$: constant and $\nu$: constant

Permalink:

http://dlmf.nist.gov/15.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(ii), §15.11 and Ch.15

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34: Errata

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Several changes have been made to

1. make consistent use of the Fourier transform notations $ℱ\left(f\right)$, $ℱ\left(\varphi \right)$ and $ℱ\left(u\right)$ where $f$ is a function of one real variable, $\varphi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. introduce the partial differential operator $𝐃$ in (1.16.30);

3. clarify the definition (1.16.32) of the partial differential operator $P(𝐃)$; and

4. clarify the use of $P(𝐃)$ and $P(𝐱)$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

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35: 21.9 Integrable Equations

… ►Here $x$ and $y$ are spatial variables, $t$ is time, and $u(x,y,t)$ is the elevation of the surface wave. All quantities are made dimensionless by a suitable scaling transformation. …

36: 16.5 Integral Representations and Integrals

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16.5.2 $${}_{p+1}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_0,\mathrm{\dots },b_q};z)=\frac{\mathrm{\Gamma }\left(b_0\right)}{\mathrm{\Gamma }\left(a_0\right)\mathrm{\Gamma }\left(b_0-a_0\right)}{\int }_0^1{t}^{a_0-1}{(1-t)}^{b_0-a_0-1}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};zt)dt,$$ $\mathrm{\Re }{b}_0>\mathrm{\Re }{a}_0>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Keywords:

Mellin transform

Referenced by:

§16.5, §16.5

Permalink:

http://dlmf.nist.gov/16.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

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16.5.3 $${}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{1}{\mathrm{\Gamma }\left(a_0\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a_0-1}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\cdots },b_q};zt)dt,$$ , $\mathrm{\Re }{a}_0>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Keywords:

Laplace transform, Mellin transform

Referenced by:

§16.5

Permalink:

http://dlmf.nist.gov/16.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

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16.5.4 $${}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_0,\mathrm{\dots },b_q};z)=\frac{\mathrm{\Gamma }\left(b_0\right)}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^t{t}^{-b_0}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\frac{z}{t})dt,$$ $c>0$, $\mathrm{\Re }{b}_0>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Keywords:

Mellin transform

Referenced by:

§16.5, §16.5

Permalink:

http://dlmf.nist.gov/16.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

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37: Bibliography T

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

9th item.

Encodings:

BibTeX

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

BibTeX

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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A note on the real zeros of the incomplete gamma function. Integral Transforms Spec. Funct. 23 (6), pp. 445–453.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Feng Qi), ZentralBlatt

Referenced by:

§8.13(i), §8.13(i).

Encodings:

BibTeX

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Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.

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External Links:

ISSN 0002-9947, Document, FullText, MathReview (John M. Voight), ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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38: 8.19 Generalized Exponential Integral

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8.19.2 $$E_p\left(z\right)=z^{p-1}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t^p}dt.$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§8.19(i), §8.19(viii)

Permalink:

http://dlmf.nist.gov/8.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19 and Ch.8

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8.19.3 $$E_p\left(z\right)={\int }_1^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^p}dt,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

A&S Ref:

5.1.4

Referenced by:

§8.19(iii)

Permalink:

http://dlmf.nist.gov/8.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

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8.19.4 $$E_p\left(z\right)=\frac{z^{p-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{p-1}{\mathrm{e}}^{-zt}}{1+t}dt,$$ , $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/8.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

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39: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

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10.46.1 $$\varphi (\rho ,\beta ;z)=\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{k!\mathrm{\Gamma }\left(\rho k+\beta \right)},$$ $\rho >-1$.

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Defines:

$\varphi (\rho ,\beta ;z)$: generalized Bessel function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.46.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.46 and Ch.10

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10.46.2 $$I_{\nu }\left(z\right)={\left(\frac{1}{2}z\right)}^{\nu }\varphi (1,\nu +1;\frac{1}{4}{z}^2).$$

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Symbols:

$\varphi (\rho ,\beta ;z)$: generalized Bessel function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.46.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.46 and Ch.10

… ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: ►

10.46.3 $$E_{a,b}\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(ak+b\right)},$$ $a>0$.

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Defines:

$E_{a,b}\left(z\right)$: Mittag-Leffler function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.46.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.46 and Ch.10

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40: 7.7 Integral Representations

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7.7.4 $${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{\sqrt{t+z^2}}dt=\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{a{z}^2}\mathrm{erfc}\left(\sqrt{a}z\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.

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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, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform

A&S Ref:

7.4.8

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.13 $$\mathrm{f}\left(z\right)=\frac{{(2\pi )}^{-3/2}}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\zeta }^{-s}\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\mathrm{\Gamma }\left(s+\frac{3}{4}\right)\mathrm{\Gamma }\left(\frac{1}{4}-s\right)ds,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\zeta$: change of variable and $c$: constant

Keywords:

Mellin transform

Referenced by:

§7.7(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.14 $$\mathrm{g}\left(z\right)=\frac{{(2\pi )}^{-3/2}}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\zeta }^{-s}\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\mathrm{\Gamma }\left(s+\frac{1}{4}\right)\mathrm{\Gamma }\left(\frac{3}{4}-s\right)ds.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\zeta$: change of variable and $c$: constant

Keywords:

Mellin transform

Referenced by:

§7.7(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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