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41: 30.10 Series and Integrals

§30.10 Series and Integrals

►Integrals and integral equations for ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …

42: 7.16 Generalized Error Functions

§7.16 Generalized Error Functions

►Generalizations of the error function and Dawson’s integral are ${\int }_0^x{\mathrm{e}}^{-t^p}dt$ and ${\int }_0^x{\mathrm{e}}^{t^p}dt$. …

43: 16.20 Integrals and Series

§16.20 Integrals and Series

►Integrals of the Meijer $G$-function are given in Apelblat (1983, §19), Erdélyi et al. (1953a, §5.5.2), Erdélyi et al. (1954a, §§6.9 and 7.5), Luke (1969a, §3.6), Luke (1975, §5.6), Mathai (1993, §3.10), and Prudnikov et al. (1990, §2.24). …

44: 19.28 Integrals of Elliptic Integrals

§19.28 Integrals of Elliptic Integrals

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19.28.5 $${\int }_z^{\mathrm{\infty }}{R}_D(x,y,t)dt=6R_F(x,y,z),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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19.28.8 $${\int }_0^{\mathrm{\infty }}{R}_J(tx,y,z,tp)dt=\frac{6}{\sqrt{p}}{R}_C(p,x)R_F(0,y,z).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

… ►To replace a single component of $𝐳$ in $R_{-a}(𝐛;𝐳)$ by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)).

45: 19.17 Graphics

§19.17 Graphics

►See Figures 19.17.1–19.17.8 for symmetric elliptic integrals with real arguments. … ►The cases $x=0$ or $y=0$ correspond to the complete integrals. … ►To view $R_F(0,y,1)$ and $2R_G(0,y,1)$ for complex $y$, put $y=1-k^2$, use (19.25.1), and see Figures 19.3.7–19.3.12. … ►

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Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. … Magnify 3D Help

46: 19.20 Special Cases

§19.20 Special Cases

… ►The general lemniscatic case is … ►where $x,y,z$ may be permuted. … ►The general lemniscatic case is … ►

47: 19.7 Connection Formulas

§19.7 Connection Formulas

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§19.7(i) Complete Integrals of the First and Second Kinds

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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Imaginary-Argument Transformation

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48: 10.64 Integral Representations

§10.64 Integral Representations

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Schläfli-Type Integrals

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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49: 7.3 Graphics

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Figure 7.3.2: Dawson’s integral $F\left(x\right)$, $-3.5\le x\le 3.5$. Magnify

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Figure 7.3.3: Fresnel integrals $C\left(x\right)$ and $S\left(x\right)$, $0\le x\le 4$. Magnify

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. Fresnel (1818) introduced the integral $ℱ\left(x\right)$ in his study of the interference pattern at the edge of a shadow. … Magnify

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50: 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

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${\mathrm{𝐸𝑐}}_{\nu }^{2m}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

… ►The first and the fourth functions have period $2\mathrm{i}{K}^{\prime }$; the second and the third have period $4\mathrm{i}{K}^{\prime }$. …