of arbitrary order

SearchAdvancedHelp

(0.002 seconds)

41—49 of 49 matching pages

41: 13.2 Definitions and Basic Properties

… ►where $\delta$ is an arbitrary small positive constant. … ►

13.2.13 $$M(a,b,z)=1+O\left(z\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

… ►

13.2.14 $$U(-n,b,z)={(-1)}^n{\left(b\right)}_n+O\left(z\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

… ►

13.2.16 $$U(a,b,z)=\frac{\mathrm{\Gamma }\left(b-1\right)}{\mathrm{\Gamma }\left(a\right)}{z}^{1-b}+O\left(z^{2-\mathrm{\Re }b}\right),$$ $\mathrm{\Re }b\ge 2$, $b\ne 2$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.5.6 (with order estimate corrected)

Permalink:

http://dlmf.nist.gov/13.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

… ►where $\delta$ is an arbitrary small positive constant. …

42: Bibliography

… ►

Polygamma functions of negative order. J. Comput. Appl. Math. 100 (2), pp. 191–199.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

(25.11.32), (25.11.34), §25.11(viii).

Encodings:

BibTeX

… ►

On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

►

Congruences of $p$-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

… ►

Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Software 26 (3), pp. 408–414.

ⓘ

Notes:

Double-precision C++, minimum accuracy 14D

External Links:

ISSN 0098-3500, Document, GAMS (module 804; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

… ►

A subroutine package for Bessel functions of a complex argument and nonnegative order. Technical Report Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.

ⓘ

External Links:

FullText

Referenced by:

1st item, B. Döring (1966).

Encodings:

BibTeX

…

43: 15.12 Asymptotic Approximations

… ►Let $\delta$ denote an arbitrary small positive constant. … ►

15.12.2 $$F(a,b;c;z)=\sum _{s=0}^{m-1}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s+O\left(c^{-m}\right),$$ $|c|\to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

Referenced by:

§15.19(iv)

Permalink:

http://dlmf.nist.gov/15.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(ii), §15.12 and Ch.15

… ►Again, throughout this subsection $\delta$ denotes an arbitrary small positive constant, and $a,b,c,z$ are real or complex and fixed. … ►

15.12.5 $$𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b-\lambda }{c};\frac{1}{2}-\frac{1}{2}z)=2^{(a+b-1)/2}\frac{{(z+1)}^{(c-a-b-1)/2}}{{(z-1)}^{c/2}}\sqrt{\zeta \sinh \zeta }{\left(\lambda +\frac{1}{2}a-\frac{1}{2}b\right)}^{1-c}\left(I_{c-1}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)(1+O({\lambda }^{-2}))+\frac{I_{c-2}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)}{2\lambda +a-b}\left(\left(c-\frac{1}{2}\right)\left(c-\frac{3}{2}\right)\left(\frac{1}{\zeta }-\coth \zeta \right)+\frac{1}{2}(2c-a-b-1)(a+b-1)\tanh \left(\frac{1}{2}\zeta \right)+O({\lambda }^{-2})\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $I_{\nu }\left(z\right)$: modified Bessel 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, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.7 $$F(\genfrac{}{}{0pt}{}{a,b-\lambda }{c+\lambda };-z)=2^{b-c+(1/2)}{\left(\frac{z+1}{2\sqrt{z}}\right)}^{\lambda }\left({\lambda }^{a/2}U(a-\frac{1}{2},-\alpha \sqrt{\lambda })\left({(1+z)}^{c-a-b}{z}^{1-c}{\left(\frac{\alpha }{z-1}\right)}^{1-a}+O({\lambda }^{-1})\right)+\frac{{\lambda }^{(a-1)/2}}{\alpha }U(a-\frac{3}{2},-\alpha \sqrt{\lambda })\left({(1+z)}^{c-a-b}{z}^{1-c}{\left(\frac{\alpha }{z-1}\right)}^{1-a}-2^{c-b-(1/2)}{\left(\frac{\alpha }{z-1}\right)}^a+O({\lambda }^{-1})\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\alpha$

Permalink:

http://dlmf.nist.gov/15.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

…

44: 2.8 Differential Equations with a Parameter

… ►For example, $u$ can be the order of a Bessel function or degree of an orthogonal polynomial. … ►(the constants of integration being arbitrary). … ►

§2.8(iv) Case III: Simple Pole

… ►For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. … ►Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007). …

45: 15.11 Riemann’s Differential Equation

… ►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by … ►for arbitrary $\lambda$ and $\mu$.

46: 13.14 Definitions and Basic Properties

… ►

13.14.14 $$M_{\kappa ,\mu }\left(z\right)=z^{\mu +\frac{1}{2}}\left(1+O\left(z\right)\right),$$ $2\mu \ne -1,-2,-3,\mathrm{\dots }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

… ►

13.14.15 $$W_{\frac{1}{2}\pm \mu +n,\mu }\left(z\right)={(-1)}^n{\left(1\pm 2\mu \right)}_n{z}^{\frac{1}{2}\pm \mu }+O\left(z^{\frac{3}{2}\pm \mu }\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

… ►

13.14.17 $$W_{\kappa ,\frac{1}{2}}\left(z\right)=\frac{1}{\mathrm{\Gamma }\left(1-\kappa \right)}+O\left(z\ln z\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

… ►

13.14.19 $$W_{\kappa ,0}\left(z\right)=-\frac{\sqrt{z}}{\mathrm{\Gamma }\left(\frac{1}{2}-\kappa \right)}\left(\ln z+\psi \left(\frac{1}{2}-\kappa \right)+2\gamma \right)+O\left(z^{3/2}\ln z\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

… ►where $\delta$ is an arbitrary small positive constant. …

47: 18.39 Applications in the Physical Sciences

… ►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ► ${ϵ}_0$ is referred to as the ground state, all others, $n=1,2,\mathrm{\dots },$ in order of increasing energy being excited states. … ►If $\mathrm{\Psi }(x,t=0)=\chi (x)$ is an arbitrary unit normalized function in the domain of $\mathscr{H}$ then, by self-adjointness, … ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials ${𝐋}_{n+l}^{2l+1}({\rho }_n)$. …

48: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►

8.18.1 $$I_x(a,b)=\mathrm{\Gamma }\left(a+b\right)x^a{(1-x)}^{b-1}\left(\sum _{k=0}^{n-1}\frac{1}{\mathrm{\Gamma }\left(a+k+1\right)\mathrm{\Gamma }\left(b-k\right)}{\left(\frac{x}{1-x}\right)}^k+O\left(\frac{1}{\mathrm{\Gamma }\left(a+n+1\right)}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(i)

Permalink:

http://dlmf.nist.gov/8.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(i), §8.18 and Ch.8

… ►

8.18.3 $$I_x(a,b)=\frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}\left(\sum _{k=0}^{n-1}{d}_k{F}_k+O\left(a^{-n}\right)F_0\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $F_k$: functions and $d_k$: coefficients

Referenced by:

Erratum (V1.0.14) for Equation (8.18.3)

Permalink:

http://dlmf.nist.gov/8.18.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Previously this equation appeared without the order estimate as $I_x(a,b)\sim \frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}{\sum }_{k=0}^{\mathrm{\infty }}{d}_k{F}_k$. The range of $x$ was extended to include $1$.

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

… ►uniformly for $x\in (0,1)$ and $a/(a+b)$, $b/(a+b)\in [\delta ,1-\delta ]$, where $\delta$ again denotes an arbitrary small positive constant. …

49: 3.8 Nonlinear Equations

… ►for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … … ►This is useful when $f(z)$ satisfies a second-order linear differential equation because of the ease of computing $f^{\prime \prime }(z_n)$. … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). … ►For an arbitrary starting point $z_0\in ℂ$, convergence cannot be predicted, and the boundary of the set of points $z_0$ that generate a sequence converging to a particular zero has a very complicated structure. …