������ ���������������������degree��������� ���kaa77788���qFHdiOS

SearchAdvancedHelp

(0.002 seconds)

31—40 of 131 matching pages

31: 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

…

32: 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

…

33: 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). …

34: 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 $n$. … ►

35: 30.7 Graphics

…

36: 14.24 Analytic Continuation

… ►

14.24.1 $$P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={\mathrm{e}}^{s\nu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right)+\frac{2\mathrm{i}\sin \left(\left(\nu +\frac{1}{2}\right)s\pi \right){\mathrm{e}}^{-s\pi \mathrm{i}/2}}{\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\mu -\nu \right)}{𝑸}_{\nu }^{\mu }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

►

14.24.2 $${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={(-1)}^s{\mathrm{e}}^{-s\nu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right),$$

ⓘ

Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.23, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

… ►

14.24.3 $$P_{\nu ,s}^{-\mu }\left(z\right)={\mathrm{e}}^{s\mu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right),$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Permalink:

http://dlmf.nist.gov/14.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

►

14.24.4 $${𝑸}_{\nu ,s}^{\mu }\left(z\right)={\mathrm{e}}^{-s\mu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right)-\frac{\pi \mathrm{i}\sin \left(s\mu \pi \right)}{\sin \left(\mu \pi \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{P}_{\nu }^{-\mu }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

…

37: 30.1 Special Notation

… ► ►►►

$x$	real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, .
…
$n$	degree, an integer $n=m,m+1,m+2,\mathrm{\dots }$.
…

…

38: William P. Reinhardt

… ►His undergraduate and graduate degrees are from the University of California at Berkeley and Harvard University, respectively. …

39: 14.3 Definitions and Hypergeometric Representations

… ►

14.3.1

ⓘ

Defines:

: Ferrers function of the first kind

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

A&S Ref:

8.1.2 (modified)

Referenced by:

§14.11, §14.15(i), §14.3(i), §14.3(i), §15.9(iv), §15.9(iv)

Permalink:

http://dlmf.nist.gov/14.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

… ►

14.3.6

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre 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, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.21(i), §14.21(iii), §14.3(ii), §14.3(iv), §15.9(iv)

Permalink:

http://dlmf.nist.gov/14.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

… ►

14.3.9

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre 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, $\mu$: general order, $\nu$: general degree and : real variable

Permalink:

http://dlmf.nist.gov/14.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

… ►

14.3.11

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree, : real variable and $w_2$: function $w_1$,

A&S Ref:

8.1.4 (modified)

Referenced by:

§10.59, §14.23, §14.3(iii), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

… ►

14.3.15

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre 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, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.3.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

…

40: 14.25 Integral Representations

… ►

14.25.1 $$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

►

14.25.2 $${𝑸}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

…