integer degree and order

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1: 30.8 Expansions in Series of Ferrers Functions

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

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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

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30.8.8 $$\frac{{\lambda }_n^m\left({\gamma }^2\right)-B_k}{A_k}\frac{a_{n,k}^m({\gamma }^2)}{a_{n,k-1}^m({\gamma }^2)}=1+O\left(\frac{1}{k^4}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients, $A_k$ and $B_k$

Permalink:

http://dlmf.nist.gov/30.8.E8

Encodings:

pMML, png, TeX

See also:

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

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2: 30.1 Special Notation

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$x$	real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, .
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$m$	order, a nonnegative integer.
$n$	degree, an integer $n=m,m+1,m+2,\mathrm{\dots }$.
$k$	integer.
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3: 30.16 Methods of Computation

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30.16.4 $${\alpha }_{p,d}-{\lambda }_n^m\left({\gamma }^2\right)=O\left(\frac{{\gamma }^{4d}}{4^{2d+1}{((m+2d-1)!(m+2d+1)!)}^2}\right),$$ $d\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\alpha }_{j,k}$: real eigenvalues and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(i), §30.16 and Ch.30

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4: 14.7 Integer Degree and Order

§14.7 Integer Degree and Order

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5: 30.11 Radial Spheroidal Wave Functions

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30.11.6

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\psi }_k^{(j)}(z)$: spherical function and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(iii), §30.11 and Ch.30

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6: 30.3 Eigenvalues

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30.3.2 $${\lambda }_n^m\left({\gamma }^2\right)=n(n+1)-\frac{1}{2}{\gamma }^2+O\left(n^{-2}\right),$$ $n\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(ii), §30.3 and Ch.30

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7: 14.21 Definitions and Basic Properties

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14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.21(i), §14.21 and Ch.14

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§14.21(iii) Properties

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8: Bibliography S

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On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

9: 10.19 Asymptotic Expansions for Large Order

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10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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10: 14.6 Integer Order

§14.6 Integer Order

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§14.6(i) Nonnegative Integer Orders

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§14.6(ii) Negative Integer Orders

… ►For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). …