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11: 28 Mathieu Functions and Hill’s Equation

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12: 36.9 Integral Identities

§36.9 Integral Identities

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36.9.1 $${|{\mathrm{\Psi }}_1\left(x\right)|}^2=2^{5/3}{\int }_0^{\mathrm{\infty }}{\mathrm{\Psi }}_1\left(2^{2/3}(3u^2+x)\right)du;$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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36.9.2 $${(\mathrm{Ai}\left(x\right))}^2=\frac{2^{2/3}}{\pi }{\int }_0^{\mathrm{\infty }}\mathrm{Ai}\left(2^{2/3}(u^2+x)\right)du.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $x$: real parameter

Referenced by:

§9.5(i)

Permalink:

http://dlmf.nist.gov/36.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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36.9.4 $${|{\mathrm{\Psi }}_2\left(x,y\right)|}^2={\int }_0^{\mathrm{\infty }}\left({\mathrm{\Psi }}_1\left(\frac{4u^3+2uy+x}{u^{1/3}}\right)+{\mathrm{\Psi }}_1\left(\frac{4u^3+2uy-x}{u^{1/3}}\right)\right)\frac{du}{u^{1/3}}.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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36.9.7 $${|{\mathrm{\Psi }}_3\left(x,y,z\right)|}^2=\frac{2^{7/4}}{5^{1/4}}{\int }_0^{\mathrm{\infty }}\mathrm{\Re }\left({\mathrm{e}}^{2\mathrm{i}u(u^4+z{u}^2+x)}{\mathrm{\Psi }}_2\left(\frac{2^{7/4}}{5^{1/4}}y{u}^{3/4},\sqrt{\frac{2u}{5}}(3z+10u^2)\right)\right)\frac{du}{u^{1/4}}.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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13: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

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36.8.3 $$\frac{3^{2/3}}{4{\pi }^2}{\mathrm{\Psi }}^{(\mathrm{H})}\left(3^{1/3}𝐱\right)=\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=0}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)c_n(y)}{n!}+\mathrm{Ai}\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)d_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)c_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=1}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)d_n(y)}{n!},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

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36.8.4 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2{\pi }^2{\left(\frac{2}{3}\right)}^{2/3}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\mathrm{i}{(2/3)}^{2/3}z\right)}^n}{n!}\mathrm{\Re }\left(f_n(\frac{x+\mathrm{i}y}{{12}^{1/3}},\frac{x-\mathrm{i}y}{{12}^{1/3}})\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter and $f_n$: function

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

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14: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

15: 23 Weierstrass Elliptic and Modular
Functions

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16: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

17: 36 Integrals with Coalescing Saddles

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18: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

19: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

20: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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