asymptotic and order symbols

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1: 2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols

… ►As $x\to c$ in $\mathbf{X}$ … ►

§2.1(ii) Integration and Differentiation

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2.1.12

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Symbols:

$\sim$: asymptotic equality, $dx$: differential, $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $f(x)$: continuous function and $\nu$: complex constant

Permalink:

http://dlmf.nist.gov/2.1.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(ii), §2.1 and Ch.2

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2: 13.9 Zeros

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13.9.16 $$a=-n-\frac{2}{\pi }\sqrt{zn}-\frac{2z}{{\pi }^2}+\frac{1}{2}b+\frac{1}{4}+\frac{z^2\left(\frac{1}{3}-4{\pi }^{-2}\right)+z-{(b-1)}^2+\frac{1}{4}}{4\pi \sqrt{zn}}+O\left(\frac{1}{n}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii), § ‣ Recent News, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16)

Permalink:

http://dlmf.nist.gov/13.9.E16

Encodings:

pMML, png, TeX

Errata (effective with 1.0.13):

In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.

Reported 2016-09-12 by Adri Olde Daalhuis

Errata (effective with 1.0.12):

Originally this equation was expressed in terms of the asymptoticsymbol $\sim$. As a consequence of the use of the $O$ ordersymbol on the right hand side, $\sim$ was replaced by $=$.

Reported 2016-07-11 by Rudi Weikard

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

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3: 18.15 Asymptotic Approximations

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18.15.22 $$L_n^{(\alpha )}\left(\nu x\right)={(-1)}^n\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha -\frac{1}{2}}{x}^{\frac{1}{2}\alpha +\frac{1}{4}}}{\left(\frac{\zeta }{x-1}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{m=0}^{M-1}\frac{E_m(\zeta )}{{\nu }^{2m}}+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{m=0}^{M-1}\frac{F_m(\zeta )}{{\nu }^{2m}}+\mathrm{envAi}\left({\nu }^{\frac{2}{3}}\zeta \right)O\left(\frac{1}{{\nu }^{2M-\frac{2}{3}}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\sim$: Poincaré asymptotic expansion, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ , $\zeta$ , $E_m$: coefficient, $F_m$: coefficient and $x$: real variable

Referenced by:

Erratum (V1.0.11) for Equation (18.15.22)

Permalink:

http://dlmf.nist.gov/18.15.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally this equation was expressed in terms of the asymptoticsymbol $\sim$. As a consequence of the use of the $O$ ordersymbol on the right-hand side, $\sim$ was replaced by $=$.

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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4: Errata

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Originally was expressed in term of asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

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Because of the use of the $O$ order symbol on the right-hand side, the asymptotic expansion for the generalized Laguerre polynomial $L_n^{(\alpha )}\left(\nu x\right)$ was rewritten as an equality.

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5: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.11 $${\mathbf{A}}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\mathrm{e}}^{-\nu \mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_k{b}_k(\lambda )}{{\nu }^k},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $b_k(\lambda )$: expansion function

Sources:

Dingle (1973, p. 388); Olver (1997b, pp. 103 and 352)

Source:

Derivable by applying Laplace’s method to the integral (11.10.4).

Permalink:

http://dlmf.nist.gov/11.11.E11

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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6: Bibliography K

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Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.

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External Links:

ISSN 0098-3500, Document, MathReview Entry

Referenced by:

1st item.

Encodings:

BibTeX

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Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument. ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.

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External Links:

Document

Referenced by:

2nd item, §10.77(viii).

Encodings:

BibTeX

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Some completely monotonic functions of positive order. Math. Comp. 79 (271), pp. 1697–1707.

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External Links:

ISSN 0025-5718, Document, FullText, MathReview (Necdet Batir), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

Encodings:

BibTeX

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An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1), pp. 3–43.

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External Links:

ISSN 0747-7171, MathReview, ZentralBlatt

Referenced by:

§31.14(ii).

Encodings:

BibTeX

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Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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Referenced by:

§36.14(i).

Encodings:

BibTeX

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7: Bibliography W

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Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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External Links:

ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

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Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

BibTeX

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Asymptotic expansions for second-order linear difference equations. II. Stud. Appl. Math. 87 (4), pp. 289–324.

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External Links:

ISSN 0022-2526, MathReview (Shao Zhu Chen), ZentralBlatt

Referenced by:

§2.9(ii).

Encodings:

BibTeX

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Asymptotic expansions for second-order linear difference equations. J. Comput. Appl. Math. 41 (1-2), pp. 65–94.

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External Links:

ISSN 0377-0427, Document, MathReview (Shao Zhu Chen), ZentralBlatt

Referenced by:

§2.9(i), §2.9(ii).

Encodings:

BibTeX

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Asymptotic solutions of a fourth order differential equation. Stud. Appl. Math. 118 (2), pp. 133–152.

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External Links:

ISSN 0022-2526, Document, MathReview Entry

Referenced by:

§2.8(vi).

Encodings:

BibTeX

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8: 2.11 Remainder Terms; Stokes Phenomenon

… ► ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable $x$ that is intended to be used. … ►For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). ►For higher-order differential equations, see Olde Daalhuis (1998a, b). … ► …

9: Bibliography T

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The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

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Referenced by:

§2.4(vi).

Encodings:

BibTeX

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Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41 (1), pp. 114–126.

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External Links:

ISSN 0044-2275, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§18.11(i), §18.16(vi).

Encodings:

BibTeX

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Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§8.12, §8.12.

Encodings:

BibTeX

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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.1, Notations E, Notations E.

Encodings:

BibTeX

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10: 14.8 Behavior at Singularities

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14.8.1 $${\mathsf{P}}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

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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

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14.8.2 $${\mathsf{P}}_{\nu }^m\left(x\right)\sim {(-1)}^m\frac{{\left(\nu -m+1\right)}_{2m}}{m!}{\left(\frac{1-x}{2}\right)}^{m/2},$$ $m=1,2,3,\mathrm{\dots }$, $\nu \ne m-1,m-2,\mathrm{\dots },-m$,

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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

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14.8.4 $${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{2}\cos \left(\mu \pi \right)\mathrm{\Gamma }\left(\mu \right){\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$,

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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

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14.8.7 $$P_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

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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

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14.8.11 $${\mathit{Q}}_{\nu }^{\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(\mu \right)}{2\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$ $\mathrm{\Re }\mu >0$, $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

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