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11: 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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16.13.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_m{\left({\alpha }^{\prime }\right)}_n{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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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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12: 31.8 Solutions via Quadratures

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31.8.3 $$g=\frac{1}{2}\max (2\underset{0\le k\le 3}{\max }{m}_k,1+N-(1+{(-1)}^N)\left(\frac{1}{2}+\underset{0\le k\le 3}{\min }{m}_k\right)).$$

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Symbols:

$m$: nonnegative integer, $g$: degree of $\lambda$ and $N$: degree of $z$

Permalink:

http://dlmf.nist.gov/31.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.8 and Ch.31

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13: Bibliography

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Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.

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Notes:

Double-precision Fortran, maximum accuracy 13D.

External Links:

ISSN 0098-3500, Document, GAMS (module 724; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Algorithm 39: Areas under the normal curve. The Computer Journal 12 (2), pp. 197–198.

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Notes:

Includes Algol program, maximum accuracy 12D.

External Links:

Document

Referenced by:

§7.25(ii).

Encodings:

BibTeX

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Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function. ACM Trans. Math. Software 9 (4), pp. 494–502.

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Notes:

Single and double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 610; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Algorithm 556: Exponential integrals. ACM Trans. Math. Software 6 (3), pp. 420–428.

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Notes:

Single-precision Fortran, maximum accuracy 14S. For a Remark see same journal v. 9 (1983), no. 4, p. 525.

External Links:

ISSN 0098-3500, Document, Remark,GAMS (module 556; package TOMS), ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 588; package TOMS)

Referenced by:

2nd item.

Encodings:

BibTeX

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14: 11.15 Approximations

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Newman (1984) gives polynomial approximations for ${𝐇}_n\left(x\right)$ for $n=0,1$, $0\le x\le 3$, and rational-fraction approximations for ${𝐇}_n\left(x\right)-Y_n\left(x\right)$ for $n=0,1$, $x\ge 3$. The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

15: 16.21 Differential Equation

… ►This equation is of order $\max (p,q)$. …

16: 30.15 Signal Analysis

… ►The maximum (or least upper bound) $\mathrm{B}$ of all numbers … ►

30.15.11 $$\arccos \sqrt{\mathrm{B}}+\arccos \sqrt{\alpha }=\arccos \sqrt{{\mathrm{\Lambda }}_0},$$

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Symbols:

$\arccos z$: arccosine function, $\mathrm{\Lambda }$ and $\mathrm{B}$: maximum bound

Permalink:

http://dlmf.nist.gov/30.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(v), §30.15 and Ch.30

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30.15.12 $$\mathrm{B}={\left(\sqrt{{\mathrm{\Lambda }}_0\alpha }+\sqrt{1-{\mathrm{\Lambda }}_0}\sqrt{1-\alpha }\right)}^2.$$

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Symbols:

$\mathrm{\Lambda }$ and $\mathrm{B}$: maximum bound

Permalink:

http://dlmf.nist.gov/30.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(v), §30.15 and Ch.30

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17: Bibliography G

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Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

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Notes:

For certification see same journal 8(1965), pp. 105–106, for remark see ACM Trans. Math. Software, 1(1975), pp. 282–284. Includes Algol program, maximum accuracy: 11D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(iii).

Encodings:

BibTeX

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Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iii).

Encodings:

BibTeX

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

BibTeX

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Algorithm 521: Repeated integrals of the coerror function. ACM Trans. Math. Software 3, pp. 301–302.

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Notes:

Single-precision Fortran, maximum accuracy 14D

External Links:

ISSN 0098-3500, Document, GAMS (module 521; package TOMS)

Referenced by:

3rd item, W. Gautschi (1977a).

Encodings:

BibTeX

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Evaluation of Legendre functions of argument greater than one. Comput. Phys. Comm. 105 (2-3), pp. 273–283.

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Notes:

Double-precision Fortran, maximum accuracy 18S

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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18: 3.2 Linear Algebra

… ►The $p$-norm of a vector $𝐱={[x_1,\mathrm{\dots },x_n]}^{\mathrm{T}}$ is given by … ►

$${\Vert 𝐱\Vert }_{\mathrm{\infty }}=\underset{1\le j\le n}{\max }\left|x_j\right|.$$

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3.2.14 $${\Vert 𝐀\Vert }_p=\underset{𝐱\ne \mathrm{𝟎}}{\max }\frac{{\Vert 𝐀𝐱\Vert }_p}{{\Vert 𝐱\Vert }_p}.$$

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Referenced by:

§3.2(iii)

Permalink:

http://dlmf.nist.gov/3.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §3.2(iii), §3.2 and Ch.3

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$${\Vert 𝐀\Vert }_1=\underset{1\le k\le n}{\max }\sum _{j=1}^n\left|a_{jk}\right|,$$

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$${\Vert 𝐀\Vert }_{\mathrm{\infty }}=\underset{1\le j\le n}{\max }\sum _{k=1}^n\left|a_{jk}\right|,$$

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19: 13.10 Integrals

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13.10.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;k/z),$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}U(a,c,t)dt=\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(b-c+1\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;a+b-c+1;1-\frac{1}{z}),$$ $\mathrm{\Re }b>\max (\mathrm{\Re }c-1,0)$, $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.11 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}U(a,b,t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)\mathrm{\Gamma }\left(\lambda -b+1\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)},$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iii), §13.10 and Ch.13

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13.10.15 $${\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -b+2\right)}{\mathrm{\Gamma }\left(a\right)}{x}^{\frac{1}{2}\nu }U(\nu -b+2,\nu -a+2,x),$$ $x>0$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

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13.10.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\mathrm{\Gamma }\left(\nu -b+2\right)x^{\frac{1}{2}\nu }{\mathrm{e}}^{-x}𝐌(a,a-b+\nu +2,x),$$ $x>0$, .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

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20: 1.4 Calculus of One Variable

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Maxima and Minima

… ►as $\max (x_{j+1}-x_j)\to 0$. … ►

§1.4(vii) Maxima and Minima

►If $f(x)$ is twice-differentiable, and if also $f^{\prime }(x_0)=0$ and ($>0$), then $x=x_0$ is a local maximum (minimum) (§1.4(iii)) of $f(x)$. The overall maximum (minimum) of $f(x)$ on $[a,b]$ will either be at a local maximum (minimum) or at one of the end points $a$ or $b$. …