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31: About Color Map

… ►The conventional CMYK color wheel (not to be confused with the traditional Artist’s color wheel) places the additive colors (red, green, blue) and the subtractive colors (yellow, cyan, magenta) at multiples of 60 degrees. In particular, the colors at 90 and 180 degrees are some vague greenish and purplish hues. … ►Specifically, by scaling the phase angle in $[0,2\pi )$ to $q$ in the interval $[0,4)$, the hue (in degrees) is computed as …

32: 18.2 General Orthogonal Polynomials

… ►A system (or set) of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, where $p_n(x)$ has degree $n$ as in §18.1(i), is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\ge 0$) if … ►If the polynomials $p_n(x)$ ($n=0,1,\mathrm{\dots },N$) are orthogonal on a finite set $X$ of $N+1$ distinct points as in (18.2.3), then the polynomial $p_{N+1}(x)$ of degree $N+1$, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on $X$. … ►

Degree lowering and raising differentiation formulas and structure relations

… ►If $A_n(x)$ and $B_n(x)$ are polynomials of degree independent of $n$, and moreover ${\pi }_n(x)$ is a polynomial $\pi (x)$ independent of $n$ then … ►Polynomials $p_n(x)$ of degree $n$ ($n=0,1,2,\mathrm{\dots }$) are called Sheffer polynomials if they are generated by a generating function of the form …

33: 30.6 Functions of Complex Argument

… ►

30.6.3 $$𝒲\left\{{\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)\right\}=\frac{{(-1)}^m(n+m)!}{(1-z^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

… ►

30.6.4 $${\mathrm{𝑃𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

►

30.6.5 $${\mathrm{𝑄𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m\left({\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\mp \frac{1}{2}\mathrm{i}\pi {\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

…

34: 30.8 Expansions in Series of Ferrers Functions

… ►

30.8.4 $$A_k{f}_{k-1}+\left(B_k-{\lambda }_n^m\left({\gamma }^2\right)\right)f_k+C_k{f}_{k+1}=0,$$

ⓘ

Symbols:

${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $A_k$ , $B_k$ and $C_k$

Referenced by:

§30.16(ii), §30.8(ii)

Permalink:

http://dlmf.nist.gov/30.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

… ►

30.8.5 $$\sum _{k=-R}^{\mathrm{\infty }}{a}_{n,k}^m({\gamma }^2)a_{n,k}^{-m}({\gamma }^2)\frac{1}{2n+4k+1}=\frac{1}{2n+1},$$

ⓘ

Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.16(ii)

Permalink:

http://dlmf.nist.gov/30.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

… ►

30.8.6 $$a_{n,k}^{-m}({\gamma }^2)=\frac{(n-m)!(n+m+2k)!}{(n+m)!(n-m+2k)!}{a}_{n,k}^m({\gamma }^2).$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.11(i)

Permalink:

http://dlmf.nist.gov/30.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

… ►

30.8.7 $$\frac{k^2{a}_{n,k}^m({\gamma }^2)}{a_{n,k-1}^m({\gamma }^2)}=\frac{{\gamma }^2}{16}+O\left(\frac{1}{k}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Permalink:

http://dlmf.nist.gov/30.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

… ►

30.8.11

ⓘ

Defines:

$C^{\prime }$ (locally)

Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(ii), §30.8 and Ch.30

…

35: 14.8 Behavior at Singularities

… ►

14.8.1 $${𝖯}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.30(ii), §14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

►

14.8.2 $${𝖯}_{\nu }^m\left(x\right)\sim {(-1)}^m\frac{{\left(\nu -m+1\right)}_{2m}}{m!}{\left(\frac{1-x}{2}\right)}^{m/2},$$ $m=1,2,3,\mathrm{\dots }$, $\nu \ne m-1,m-2,\mathrm{\dots },-m$,

ⓘ

Symbols:

: Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.30(ii), §14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

… ►

14.8.4 $${𝖰}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{2}\cos \left(\mu \pi \right)\mathrm{\Gamma }\left(\mu \right){\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

… ►

14.8.7 $$P_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii), §14.24, §14.8(ii)

Permalink:

http://dlmf.nist.gov/14.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

►

14.8.8 $$P_{\nu }^m\left(x\right)\sim \frac{\mathrm{\Gamma }\left(\nu +m+1\right)}{m!\mathrm{\Gamma }\left(\nu -m+1\right)}{\left(\frac{x-1}{2}\right)}^{m/2},$$ $m=1,2,3,\mathrm{\dots }$, $\nu \pm m\ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

…

36: 24.17 Mathematical Applications

… ►are called Euler splines of degree $n$. … ►A function of the form $x^n-S(x)$, with $S(x)\in {𝒮}_{n-1}$ is called a cardinal monospline of degree $n$. …$M_n(x)$ is a monospline of degree $n$, and it follows from (24.4.25) and (24.4.27) that …For each $n=1,2,\mathrm{\dots }$ the function $M_n(x)$ is also the unique cardinal monospline of degree $n$ satisfying (24.17.6), provided that … ►is the unique cardinal monospline of degree $n$ having the least supremum norm ${\Vert F\Vert }_{\mathrm{\infty }}$ on $ℝ$ (minimality property). …

37: Bibliography H

… ►

Algorithm 55: Complete elliptic integral of the first kind. Comm. ACM 4 (4), pp. 180.

ⓘ

Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 6 (1966), pp. 166–167

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

►

Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

ⓘ

Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

…

38: Bibliography W

… ►

Rational function certification of multisum/integral/“$q$” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.

ⓘ

External Links:

ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

…

39: 14.7 Integer Degree and Order

§14.7 Integer Degree and Order

… ►where $P_n\left(x\right)$ is the Legendre polynomial of degree $n$. … ►When $m$ is even and $m\le n$, ${𝖯}_n^m\left(x\right)$ and $P_n^m\left(x\right)$ are polynomials of degree $n$. … ►

40: 30.7 Graphics

…