of arbitrary order

SearchAdvancedHelp

(0.002 seconds)

11—20 of 49 matching pages

11: 11.9 Lommel Functions

… ►

11.9.2 $$w=s_{\mu ,\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{Y}_{\nu }\left(z\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/11.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

… ►

11.9.9 $$S_{\mu ,\nu }\left(z\right)\sim z^{\mu -1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{a}_k(-\mu ,\nu )z^{-2k},$$ $z\to \mathrm{\infty }$, .

ⓘ

Symbols:

$S_{\mu ,\nu }\left(z\right)$: Lommel function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant and $a_k(\mu ,\nu )$: expansion function

Referenced by:

§11.9(iii), §11.9(iii)

Permalink:

http://dlmf.nist.gov/11.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(iii), §11.9 and Ch.11

…

12: 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

…

13: 11.1 Special Notation

§11.1 Special Notation

… ► ►►►►►

$x$	real variable.
…
$\nu$	real or complex order.
$n$	integer order.
…
$\delta$	arbitrary small positive constant.

…

14: 10.1 Special Notation

… ► ►►►

$m,n$	integers. In §§10.47–10.71 $n$ is nonnegative.
…
$\delta$	arbitrary small positive constant.
…

… ►For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

15: Bibliography O

… ►

Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.

ⓘ

External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

… ►

Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

ⓘ

External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

…

16: 3.6 Linear Difference Equations

… ►

3.6.9 $$\left|\frac{e_N}{p_N{p}_{N+1}}\right|\le ϵ\underset{1\le n\le M}{\min }\left|\frac{e_n}{p_n{p}_{n+1}}\right|.$$

ⓘ

Symbols:

$N$: arbitrary positive integer, $ϵ$: relative accuracy, $M$: order, $p_n$: solutions and $e_n$: sequence

Referenced by:

§3.6(vi)

Permalink:

http://dlmf.nist.gov/3.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(v), §3.6 and Ch.3

…

17: 24.16 Generalizations

… ►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

18: Bibliography M

… ►

Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank. Methods Appl. Anal. 4 (3), pp. 250–260.

ⓘ

External Links:

ISSN 1073-2772, MathReview (W. Balser), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

…

19: 14.15 Uniform Asymptotic Approximations

… ►

§14.15(i) Large $\mu$, Fixed $\nu$

►For the interval with fixed $\nu$, real $\mu$, and arbitrary fixed values of the nonnegative integer $J$, … ►In this and subsequent subsections $\delta$ denotes an arbitrary constant such that . … ►

14.15.5

ⓘ

Symbols:

$\mu$: general order, $\nu$: general degree and $\alpha$

Referenced by:

§14.15(ii)

Permalink:

http://dlmf.nist.gov/14.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

… ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). …

20: 14.1 Special Notation

§14.1 Special Notation

… ► ►►►►►

$x$, $y$, $\tau$	real variables.
…
$m$, $n$	unless stated otherwise, nonnegative integers, used for order and degree, respectively.
$\mu$, $\nu$	general order and degree, respectively.
…
$\delta$	arbitrary small positive constant.
…

…