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11—20 of 131 matching pages

11: 31.8 Solutions via Quadratures

… ►

31.8.2 $$w_{\pm }(𝐦;\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.8 and Ch.31

… ►Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. The degree $g$ is given by ►

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

ⓘ

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

…

12: 14.31 Other Applications

… ►

§14.31(iii) Miscellaneous

… ►Legendre functions $P_{\nu }\left(x\right)$ of complex degree $\nu$ appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …

13: 14.11 Derivatives with Respect to Degree or Order

§14.11 Derivatives with Respect to Degree or Order

►

14.11.1 $$\frac{\partial }{\partial \nu }{𝖯}_{\nu }^{\mu }\left(x\right)=\pi \cot \left(\nu \pi \right){𝖯}_{\nu }^{\mu }\left(x\right)-\frac{1}{\pi }{𝖠}_{\nu }^{\mu }(x),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

… ►

14.11.3 $${𝖠}_{\nu }^{\mu }(x)=\sin \left(\nu \pi \right){\left(\frac{1+x}{1-x}\right)}^{\mu /2}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\frac{1}{2}x\right)}^k\mathrm{\Gamma }\left(k-\nu \right)\mathrm{\Gamma }\left(k+\nu +1\right)}{k!\mathrm{\Gamma }\left(k-\mu +1\right)}\left(\psi \left(k+\nu +1\right)-\psi \left(k-\nu \right)\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

►

14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

►

14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

ⓘ

Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

…

14: 14.14 Continued Fractions

… ►

14.14.1 $$\frac{1}{2}{\left(x^2-1\right)}^{1/2}\frac{P_{\nu }^{\mu }\left(x\right)}{P_{\nu }^{\mu -1}\left(x\right)}=\frac{x_0}{y_0+}\frac{x_1}{y_1+}\frac{x_2}{y_2+}\mathrm{\cdots },$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

… ►

14.14.3 $$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$ $\nu \ne \mu$,

ⓘ

Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

…

15: 30.5 Functions of the Second Kind

… ►

30.5.1 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$ $n=m,m+1,m+2,\mathrm{\dots }$.

ⓘ

Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: 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/30.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

… ►

30.5.2 $${\mathrm{𝖰𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m+1}{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$

ⓘ

Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: 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/30.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

… ►

30.5.3 $${\mathrm{𝖰𝗌}}_n^m(x,0)={𝖰}_n^m\left(x\right);$$

ⓘ

Symbols:

: Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree

Permalink:

http://dlmf.nist.gov/30.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

… ►

30.5.4 $$𝒲\left\{{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\right\}=\frac{(n+m)!}{(1-x^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)(\ne 0),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: 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, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

…

16: George E. Andrews

… ►He holds honorary degrees from the Universities of Parma, Florida, Waterloo, Illinois, and SASTRA University (India). …

17: Ranjan Roy

… ►(1974) degrees in mathematics from the Indian Institute of Technology Kharagpur, Indian Institute of Technology Kanpur, and the State University of New York at Stony Brook, respectively. …

18: 14.10 Recurrence Relations and Derivatives

… ►

14.10.1 $${𝖯}_{\nu }^{\mu +2}\left(x\right)+2(\mu +1)x{\left(1-x^2\right)}^{-1/2}{𝖯}_{\nu }^{\mu +1}\left(x\right)+(\nu -\mu )(\nu +\mu +1){𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.2 $${\left(1-x^2\right)}^{1/2}{𝖯}_{\nu }^{\mu +1}\left(x\right)-(\nu -\mu +1){𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1)x{𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.3 $$(\nu -\mu +2){𝖯}_{\nu +2}^{\mu }\left(x\right)-(2\nu +3)x{𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1){𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.3 (modified)

Referenced by:

§14.10, §14.14, §14.21(iii)

Permalink:

http://dlmf.nist.gov/14.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.4 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\mu -\nu -1){𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +1)x{𝖯}_{\nu }^{\mu }\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.5(i)

Permalink:

http://dlmf.nist.gov/14.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.5 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\nu +\mu ){𝖯}_{\nu -1}^{\mu }\left(x\right)-\nu x{𝖯}_{\nu }^{\mu }\left(x\right).$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.4 (modified)

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

…

19: 30.4 Functions of the First Kind

… ►

30.4.1 $${\int }_{-1}^1{\left({\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right)}^{2}dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$$

ⓘ

Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

A&S Ref:

21.7.11

Referenced by:

§30.4(iii)

Permalink:

http://dlmf.nist.gov/30.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(i), §30.4 and Ch.30

… ►

30.4.2 $${\mathrm{𝖯𝗌}}_n^m(x,0)={𝖯}_n^m\left(x\right);$$

ⓘ

Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(i), §30.4 and Ch.30

… ►

30.4.3 $${\mathrm{𝖯𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m}{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2).$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: 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/30.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(ii), §30.4 and Ch.30

… ►

30.4.4 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)={(1-x^2)}^{\frac{1}{2}m}\sum _{k=0}^{\mathrm{\infty }}{g}_k{x}^k,$$ $-1\le x\le 1$,

ⓘ

Symbols:

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

A&S Ref:

21.7.18

Permalink:

http://dlmf.nist.gov/30.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iii), §30.4 and Ch.30

… ►

30.4.7 $$f(x)=\sum _{n=m}^{\mathrm{\infty }}{c}_n{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

…

20: 14.9 Connection Formulas

… ►

14.9.3 $${𝖯}_{\nu }^{-m}\left(x\right)={(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{𝖯}_{\nu }^m\left(x\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.17(iii), §14.18(ii), §14.3(i), §14.6(ii), §14.9(i)

Permalink:

http://dlmf.nist.gov/14.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(i), §14.9 and Ch.14

►

14.9.4 $${𝖰}_{\nu }^{-m}\left(x\right)={(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{𝖰}_{\nu }^m\left(x\right),$$ $\nu \ne m-1,m-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.6(ii), §14.9(i)

Permalink:

http://dlmf.nist.gov/14.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(i), §14.9 and Ch.14

… ►

14.9.13 $$P_{\nu }^{-m}\left(x\right)=\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{P}_{\nu }^m\left(x\right),$$ $\nu \ne m-1,m-2,\mathrm{\dots }$.

ⓘ

Symbols:

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

Referenced by:

§14.19(v), §14.6(ii)

Permalink:

http://dlmf.nist.gov/14.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

►

14.9.14 $${𝑸}_{\nu }^{-\mu }\left(x\right)={𝑸}_{\nu }^{\mu }\left(x\right),$$

ⓘ

Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.6 (modified)

Referenced by:

§14.23, §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

… ►

14.9.16 $${𝑸}_{\nu }^{\mu }\left(x\right)={\left(\frac{1}{2}\pi \right)}^{1/2}{\left(x^2-1\right)}^{-1/4}{P}_{-\mu -(1/2)}^{-\nu -(1/2)}\left(x{\left(x^2-1\right)}^{-1/2}\right).$$

ⓘ

Symbols:

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

A&S Ref:

8.2.7 (modified)

Referenced by:

§14.19(v), §14.21(iii), §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iv), §14.9 and Ch.14

…