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11: 18.40 Methods of Computation

… ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let $N$ be a positive integer and define … ►

18.40.2 $$a_1={\mu }_0,a_n=\frac{P_{1,n+1}}{P_{1,n}{P}_{1,n-1}},$$ $n=2,\mathrm{\dots },2N+3$,

ⓘ

Symbols:

$N$: positive integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►

18.40.3 $${\alpha }_0=a_2,{\alpha }_n=a_{2n+1}+a_{2n+2},{\beta }_n=a_{2n}{a}_{2n+1},$$ $n=1,\mathrm{\dots },N$.

ⓘ

Symbols:

$N$: positive integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. …

12: 8.17 Incomplete Beta Functions

… ►Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\le x\le 1$. … ►

8.17.4 $$I_x(a,b)=1-I_{1-x}(b,a).$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.3 (Symmetry relation)

Referenced by:

§8.17(v), §8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(i), §8.17 and Ch.8

… ►With $a>0$, $b>0$, and , … ►

8.17.13 $$(a+b)I_x(a,b)=a{I}_x(a+1,b)+b{I}_x(a,b+1),$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.7

Permalink:

http://dlmf.nist.gov/8.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(iv), §8.17 and Ch.8

… ►

8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

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Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

…

13: Bibliography F

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

ⓘ

External Links:

ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii), §15.12(iii).

Encodings:

BibTeX

… ►

ⓘ

Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

… ►

Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

… ►

Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

… ►

On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

…

14: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►

§12.10(vi) Modifications of Expansions in Elementary Functions

… ► … ►

Modified Expansions

… ►

15: 26.12 Plane Partitions

… ►A plane partition, $\pi$, of a positive integer $n$, is a partition of $n$ in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … ►

26.12.20 $$\sum _{\pi \subseteq \mathbb{N}\times \mathbb{N}\times \mathbb{N}}{q}^{|\pi |}=\prod _{k=1}^{\mathrm{\infty }}\frac{1}{{(1-q^k)}^k},$$

ⓘ

Symbols:

$\mathbb{N}$: set of all positive integers, $\subseteq$: is, or is contained in, $k$: nonnegative integer, $\pi$: plane partition and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

►

26.12.21 $$\sum _{\pi \subseteq B(r,s,t)}{q}^{|\pi |}=\prod _{(h,j,k)\in B(r,s,t)}\frac{1-q^{h+j+k-1}}{1-q^{h+j+k-2}}=\prod _{h=1}^r\prod _{j=1}^s\frac{1-q^{h+j+t-1}}{1-q^{h+j-1}},$$

ⓘ

Symbols:

$\in$: element of, $\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $k$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $s$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

►

26.12.22

ⓘ

Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Referenced by:

§26.12(ii)

Permalink:

http://dlmf.nist.gov/26.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

… ►

26.12.24

ⓘ

Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

…

16: 20.11 Generalizations and Analogs

… ►For relatively prime integers $m,n$ with $n>0$ and $mn$ even, the Gauss sum $G(m,n)$ is defined by ►

20.11.1 $$G(m,n)=\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n};$$

ⓘ

Defines:

$G(m,n)$: Gauss sum (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

… ►

20.11.3 $$f(a,b)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}^{n(n+1)/2}{b}^{n(n-1)/2},$$

ⓘ

Symbols:

$n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(ii), §20.11 and Ch.20

… ►

20.11.4 $$f(a,b)={\theta }_3\left(z\right|\tau ).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(ii), §20.11 and Ch.20

… ►

20.11.8 $${\phi }_{m,n}(z,q)=\frac{{\theta }_n(0,q){\theta }_m(z,q)}{{\theta }_m(0,q){\theta }_n(z,q)},$$ $m,n=2,3,4$.

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\phi }_{n,m}(z,q)$: combined theta function, $m$: integer, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(v), §20.11 and Ch.20

…

17: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …They tend to thin out among the large integers, but this thinning out is not completely regular. … ►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. … ►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. …is the number of $k$-tuples of integers $\le n$ whose greatest common divisor is relatively prime to $n$. …

18: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

►

§26.9(i) Definitions

… ►

§26.9(ii) Generating Functions

… ►

§26.9(iii) Recurrence Relations

… ►

§26.9(iv) Limiting Form

…

19: 25.12 Polylogarithms

… ►

25.12.1 $${\mathrm{Li}}_2\left(z\right)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^2},$$ $|z|\le 1$.

ⓘ

Defines:

${\mathrm{Li}}_2\left(z\right)$: dilogarithm

Symbols:

$\equiv$: equals by definition, $n$: nonnegative integer, $x$: real variable and $z$: complex variable

Keywords:

definition

Source:

Maximon (2003, (3.1), p. 2808)

A&S Ref:

27.7.2 (with $z=1-x$)

Referenced by:

§25.12(i), §25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

… ►

25.12.5 $${\mathrm{Li}}_2\left(z^m\right)=m\sum _{k=0}^{m-1}{\mathrm{Li}}_2\left(z{\mathrm{e}}^{2\pi \mathrm{i}k/m}\right),$$ $m=1,2,3,\mathrm{\dots }$, .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $m$: nonnegative integer and $z$: complex variable

Source:

Maximon (2003, (3.12), p. 2809)

Permalink:

http://dlmf.nist.gov/25.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

… ►

►►

Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

►

►

►

Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

… ►

25.12.10 $${\mathrm{Li}}_s\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^s}.$$

ⓘ

Defines:

${\mathrm{Li}}_s\left(z\right)$: polylogarithm

Symbols:

$n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Lewin (1981, p. 236)

Referenced by:

(25.14.3)

Permalink:

http://dlmf.nist.gov/25.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12 and Ch.25

…

20: 36.2 Catastrophes and Canonical Integrals

… ►

36.2.1 $${\mathrm{\Phi }}_K(t;𝐱)=t^{K+2}+\sum _{m=1}^K{x}_m{t}^m.$$

ⓘ

Defines:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$

Symbols:

$n$: integer, $t$: variable, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Referenced by:

§36.10(i), §36.12(i), §36.5(ii), §36.7(ii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

… ►

36.2.22 $${\mathrm{\Psi }}_{2K}\left({𝐱}^{\prime }\right)={\mathrm{\Psi }}_{2K}\left(𝐱\right),$$ $x_{2m+1}^{\prime }=-x_{2m+1}$, $x_{2m}^{\prime }=x_{2m}$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $n$: integer, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

►

36.2.23 $${\mathrm{\Psi }}_{2K+1}\left({𝐱}^{\prime }\right)=\overline{{\mathrm{\Psi }}_{2K+1}\left(𝐱\right)},$$ $x_{2m+1}^{\prime }=x_{2m+1}$, $x_{2m}^{\prime }=-x_{2m}$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\overline{z}$: complex conjugate, $n$: integer, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

… ►

36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E28

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

►

36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

ⓘ

Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36