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11: 16.15 Integral Representations and Integrals

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16.15.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)}{\int }_0^1\frac{u^{\alpha -1}{(1-u)}^{\gamma -\alpha -1}}{{(1-ux)}^{\beta }{(1-uy)}^{{\beta }^{\prime }}}du,$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\left(\gamma -\alpha \right)>0$,

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

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16.15.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-{\beta }^{\prime }\right)}{\int }_0^1{\int }_0^1\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u)}^{\gamma -\beta -1}{(1-v)}^{{\gamma }^{\prime }-{\beta }^{\prime }-1}}{{(1-ux-vy)}^{\alpha }}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

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16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

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16.15.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)}{\int }_0^1{\int }_0^1\frac{u^{\alpha -1}{v}^{\beta -1}{(1-u)}^{\gamma -\alpha -1}{(1-v)}^{{\gamma }^{\prime }-\beta -1}}{{(1-ux)}^{\gamma +{\gamma }^{\prime }-\alpha -1}{(1-vy)}^{\gamma +{\gamma }^{\prime }-\beta -1}{(1-ux-vy)}^{\alpha +\beta -\gamma -{\gamma }^{\prime }+1}}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\alpha >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }\beta >0$.

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Source:

Burchnall and Chaundy (1940, (68))

Referenced by:

§16.15

Permalink:

http://dlmf.nist.gov/16.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

…

12: 27.21 Tables

§27.21 Tables

►Lehmer (1914) lists all primes up to 100 06721. …8 gives examples of primitive roots of all primes $\le 9973$; Table 24. 9 lists all primes that are less than 1 00000. … ►

13: 16.13 Appell Functions

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16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

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

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Symbols:

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

Permalink:

http://dlmf.nist.gov/16.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_m{\left({\alpha }^{\prime }\right)}_n{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function

Symbols:

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

Permalink:

http://dlmf.nist.gov/16.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

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Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

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

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►Here and elsewhere it is assumed that neither of the bottom parameters $\gamma$ and ${\gamma }^{\prime }$ is a nonpositive integer. …

14: 16.16 Transformations of Variables

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16.16.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\beta +{\beta }^{\prime };x,y)={(1-y)}^{-\alpha }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\beta +{\beta }^{\prime }};\frac{x-y}{1-y}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

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16.16.3 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,\alpha ;x,y)={(1-y)}^{-{\beta }^{\prime }}{F}_1(\beta ;\alpha -{\beta }^{\prime },{\beta }^{\prime };\gamma ;x,\frac{x}{1-y}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Permalink:

http://dlmf.nist.gov/16.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

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16.16.4 $$F_3(\alpha ,\gamma -\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)={(1-y)}^{-{\beta }^{\prime }}{F}_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,\frac{y}{y-1}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function

Permalink:

http://dlmf.nist.gov/16.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

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16.16.8 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)={(1-x)}^{-\beta }{(1-y)}^{-{\beta }^{\prime }}{F}_1(\gamma -\alpha ;\beta ,{\beta }^{\prime };\gamma ;\frac{x}{x-1},\frac{y}{y-1})={(1-x)}^{-\alpha }{F}_1(\alpha ;\gamma -\beta -{\beta }^{\prime },{\beta }^{\prime };\gamma ;\frac{x}{x-1},\frac{y-x}{1-x}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Permalink:

http://dlmf.nist.gov/16.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(ii), §16.16 and Ch.16

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16.16.9 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)={(1-x)}^{-\alpha }{F}_2(\alpha ;\gamma -\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };\frac{x}{x-1},\frac{y}{1-x}),$$

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Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Permalink:

http://dlmf.nist.gov/16.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(ii), §16.16 and Ch.16

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15: 9.9 Zeros

… ►On the real line, $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ each have an infinite number of zeros, all of which are negative. … ► $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ have no other zeros. However, $\mathrm{Bi}\left(z\right)$ and ${\mathrm{Bi}}^{\prime }\left(z\right)$ each have an infinite number of complex zeros. … ►Tables 9.9.1 and 9.9.2 give 10D values of the first ten real zeros of $\mathrm{Ai}$, ${\mathrm{Ai}}^{\prime }$, $\mathrm{Bi}$, ${\mathrm{Bi}}^{\prime }$, together with the associated values of the derivative or the function. Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of $\mathrm{Bi}$ and ${\mathrm{Bi}}^{\prime }$ in the upper half plane. …

16: 9.20 Software

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§9.20(ii) $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, $x\in ℝ$

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§9.20(iii) $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$, $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$, $z\in ℂ$

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17: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). …

18: 27.22 Software

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GIMPS. This includes updates of the largest known Mersenne prime.

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Prime Pages. Information on primes, primality testing, and factorization including links to programs and lists of primes.

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19: 32.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. …

20: 16.1 Special Notation

… ►The main functions treated in this chapter are the generalized hypergeometric function ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$, the Appell (two-variable hypergeometric) functions $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, and the Meijer $G$-function $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$. Alternative notations are ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$, and ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ for the generalized hypergeometric function, $F_1(\alpha ,\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ,\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, for the Appell functions, and $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for the Meijer $G$-function.