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31: 5.23 Approximations

… ►See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\mathrm{\Gamma }\left(z\right)$. …

32: 7.18 Repeated Integrals of the Complementary Error Function

… ►

►►

Figure 7.18.1: Repeated integrals of the scaled complementary error function $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=0,1,2,4,8,16$. Magnify

…

33: 10.16 Relations to Other Functions

… ►With $𝐅$ as in §15.2(i), and with $z$ and $\nu$ fixed, ►

10.16.10 $$J_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim𝐅(\lambda ,\mu ;\nu +1;-z^2/(4\lambda \mu )),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.70 (modified)

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

…

34: Philip J. Davis

… ►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …

35: 19.16 Definitions

… ►

19.16.1 $$R_F(x,y,z)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)},$$

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Defines:

$R_F(x,y,z)$: symmetric elliptic integral of first kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $s(t)$: scaling factor

Referenced by:

§19.16(i), §19.16(i), §19.18(i), (19.25.35), (19.25.40), §19.25(vi)

Permalink:

http://dlmf.nist.gov/19.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

►

19.16.2 $$R_J(x,y,z,p)=\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)(t+p)},$$

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Defines:

$R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $s(t)$: scaling factor

Referenced by:

§19.16(i), §19.19, §19.20(iii), §19.28

Permalink:

http://dlmf.nist.gov/19.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

►

19.16.2_5 $$R_G(x,y,z)=\frac{1}{4}{\int }_0^{\mathrm{\infty }}\frac{1}{s(t)}\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)tdt.$$

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Defines:

$R_G(x,y,z)$: symmetric elliptic integral of second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $s(t)$: scaling factor

Source:

Carlson (1977b, (9.1-9))

Referenced by:

§19.16(i), §19.16(i), §19.21(ii), (19.23.7), (19.23.7), §19.23, Erratum (V1.1.0) for Rearrangement

Permalink:

http://dlmf.nist.gov/19.16.E2_5

Encodings:

pMML, png, TeX

Rearrangement (effective with 1.1.0):

This integral representation was moved from (19.23.7) and is taken as the definition of $R_G(x,y,z)$. We have also employed the notation $s(t)$ for the factor of radicals.

See also:

Annotations for §19.16(i), §19.16 and Ch.19

… ►

19.16.4 $$s(t)=\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}.$$

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Defines:

$s(t)$: scaling factor (locally)

Referenced by:

§19.16(i), (19.25.35), §19.25(vi)

Permalink:

http://dlmf.nist.gov/19.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

… ►

19.16.5 $$R_D(x,y,z)=R_J(x,y,z,z)=\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)(t+z)},$$

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Defines:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Symbols:

$R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ , $\int$: integral and $s(t)$: scaling factor

Referenced by:

§19.18(i), §19.19, §19.20(iii), §19.20(iv), §19.24(ii), §19.6(iv)

Permalink:

http://dlmf.nist.gov/19.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

…

36: 21.9 Integrable Equations

… ►All quantities are made dimensionless by a suitable scaling transformation. …

37: Mathematical Introduction

… ►For example, for the hypergeometric function we often use the notation $𝐅(a,b;c;z)$ (§15.2(i)) in place of the more conventional ${}_2{}^{}F_1^{}(a,b;c;z)$ or $F(a,b;c;z)$. This is because $𝐅$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $𝐅$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. …

38: 3.10 Continued Fractions

… ►However, this may be unstable; also overflow and underflow may occur when evaluating $A_n$ and $B_n$ (making it necessary to re-scale from time to time). … ►In contrast to the preceding algorithms in this subsection no scaling problems arise and no a priori information is needed. … ►Again, no scaling problems arise and no a priori information is needed. …

39: 10.22 Integrals

… ►

10.22.49 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{J}_{\nu }\left(bt\right)dt=\frac{{(\frac{1}{2}b)}^{\nu }}{a^{\mu +\nu }}\mathrm{\Gamma }\left(\mu +\nu \right)𝐅(\frac{\mu +\nu }{2},\frac{\mu +\nu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\mu +\nu \right)>0,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

►

10.22.50 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{Y}_{\nu }\left(bt\right)dt=\cot \left(\nu \pi \right)\frac{{(\frac{1}{2}b)}^{\nu }\mathrm{\Gamma }\left(\mu +\nu \right)}{{(a^2+b^2)}^{\frac{1}{2}(\mu +\nu )}}𝐅(\frac{\mu +\nu }{2},\frac{1-\mu +\nu }{2};\nu +1;\frac{b^2}{a^2+b^2})-\csc \left(\nu \pi \right)\frac{{(\frac{1}{2}b)}^{-\nu }\mathrm{\Gamma }\left(\mu -\nu \right)}{{(a^2+b^2)}^{\frac{1}{2}(\mu -\nu )}}𝐅(\frac{\mu -\nu }{2},\frac{1-\mu -\nu }{2};1-\nu ;\frac{b^2}{a^2+b^2}),$$ $\mathrm{\Re }\mu >|\mathrm{\Re }\nu |,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.22.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

►For the hypergeometric function $𝐅$ see §15.2(i). … ►

10.22.56 $${\int }_0^{\mathrm{\infty }}\frac{J_{\mu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{a^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu -\frac{1}{2}\lambda +\frac{1}{2}\right)}{2^{\lambda }{b}^{\mu -\lambda +1}\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\lambda +\frac{1}{2}\right)}𝐅(\frac{1}{2}(\mu +\nu -\lambda +1),\frac{1}{2}(\mu -\nu -\lambda +1);\mu +1;\frac{a^2}{b^2}),$$ , $\mathrm{\Re }\left(\mu +\nu +1\right)>\mathrm{\Re }\lambda >-1$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

A&S Ref:

11.4.33, 11.4.34

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E56

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

… ►

10.22.64

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $n$: integer

Referenced by:

§10.22(iv), §10.59

Permalink:

http://dlmf.nist.gov/10.22.E64

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

…

40: 16.2 Definition and Analytic Properties

… ►

16.2.5 $${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)/\left(\mathrm{\Gamma }\left(b_1\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q\right)\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k}{\mathrm{\Gamma }\left(b_1+k\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q+k\right)}\frac{z^k}{k!};$$

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Defines:

${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function

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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: 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

Referenced by:

§16.2(v)

Permalink:

http://dlmf.nist.gov/16.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(v), §16.2 and Ch.16

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