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21: 8.11 Asymptotic Approximations and Expansions

… ►For bounds on $R_n(a,z)$ when $a$ is real and $z$ is complex see Olver (1997b, pp. 109–112). … ►

8.11.18 $$S_n(x)\sim \sum _{k=0}^{\mathrm{\infty }}{d}_k(x)n^{-k},$$ $n\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $S_n(x)$: function and $d_k(x)$: coefficients

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(v), §8.11 and Ch.8

►uniformly for $x\in (-\mathrm{\infty },1-\delta ]$, with ►

$$d_0(x)=x/(1-x),$$

►

$$d_k(x)=\frac{{(-1)}^k{b}_k(x)}{{(1-x)}^{2k+1}},$$ $k=1,2,3,\mathrm{\dots }$,

…

22: Brian D. Sleeman

Profile
Brian D. Sleeman

… ►Brian D. Sleeman (b. …, d. … ►D. …D. …

23: 31.12 Confluent Forms of Heun’s Equation

… ►

31.12.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+ϵ\right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

… ►

31.12.2 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\delta }{z^2}+\frac{\gamma }{z}+1\right)\frac{dw}{dz}+\frac{\alpha z-q}{z^2}w=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

… ►

31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

Erratum (V1.0.7) for Equation (31.12.3)

Permalink:

http://dlmf.nist.gov/31.12.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek

See also:

Annotations for §31.12, §31.12 and Ch.31

… ►

31.12.4 $$\frac{d^{2}w}{{dz}^2}+\left(\gamma +z\right)z\frac{dw}{dz}+\left(\alpha z-q\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

… ►For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000). …

24: 33.14 Definitions and Basic Properties

… ►

33.14.1 $$\frac{d^{2}w}{{dr}^2}+\left(ϵ+\frac{2}{r}-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{r^2}\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Referenced by:

§33.23(iii)

Permalink:

http://dlmf.nist.gov/33.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(i), §33.14 and Ch.33

… ►

33.14.13 $${\int }_0^{\mathrm{\infty }}s({ϵ}_1,\mathrm{\ell };r)s({ϵ}_2,\mathrm{\ell };r)dr=\delta \left({ϵ}_1-{ϵ}_2\right),$$ ${ϵ}_1,{ϵ}_2>0$,

ⓘ

Symbols:

$s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.0.24):

The constraint ${ϵ}_1,{ϵ}_2>0$ was added.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

… ►

33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and ${\varphi }_{n,\mathrm{\ell }}(r)$: function

Proof sketch:

Derivable from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.24):

This equation, originally written as a definite integral ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was rewritten more generally as an orthonormality relation.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

…

25: 30.12 Generalized and Coulomb Spheroidal Functions

… ►

30.12.1 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +\alpha z+{\gamma }^2(1-z^2)-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.12

Permalink:

http://dlmf.nist.gov/30.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

… ►

30.12.2 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +{\gamma }^2(1-z^2)-\frac{\alpha (\alpha +1)}{z^2}-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Permalink:

http://dlmf.nist.gov/30.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

…

26: Bibliography C

… ►

Hadronic matter at high density. Phys. Rev. D 5 (12), pp. 3231–3242.

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External Links:

Document

Referenced by:

§5.20.

Encodings:

BibTeX

… ►

Lauricella’s hypergeometric function $F_D$ . J. Math. Anal. Appl. 7 (3), pp. 452–470.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§19.15, §19.16(iii), §19.18(ii), §19.25(vii), §19.28, §19.28, Profile Bille C. Carlson.

Encodings:

BibTeX

… ►

Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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External Links:

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

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.

Encodings:

BibTeX

… ►

Chebyshev approximations for Dawson’s integral. Math. Comp. 24 (109), pp. 171–178.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

… ►

On the Lambert $W$ function. Adv. Comput. Math. 5 (4), pp. 329–359.

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External Links:

ISSN 1019-7168, MathReview, ZentralBlatt

Referenced by:

(4.13.10), (4.13.11), (4.13.1_1), (4.13.5_1), §4.13, §4.44, §4.44.

Encodings:

BibTeX

…

27: 14.29 Generalizations

… ►

14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

ⓘ

Symbols:

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

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

…

28: 22.9 Cyclic Identities

… ► … ►

22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2).$$

ⓘ

Symbols:

$d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

§22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

… ►

22.9.11 $${\left(d_{1,2}^{(2)}\right)}^2{d}_{2,2}^{(2)}\pm {\left(d_{2,2}^{(2)}\right)}^2{d}_{1,2}^{(2)}=k^{\prime }\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$$

ⓘ

Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

… ►

22.9.17 $$d_{1,4}^{(2)}{d}_{2,4}^{(2)}{d}_{3,4}^{(2)}\pm d_{2,4}^{(2)}{d}_{3,4}^{(2)}{d}_{4,4}^{(2)}+d_{3,4}^{(2)}{d}_{4,4}^{(2)}{d}_{1,4}^{(2)}\pm d_{4,4}^{(2)}{d}_{1,4}^{(2)}{d}_{2,4}^{(2)}=k^{\prime }\left(\pm d_{1,4}^{(2)}+d_{2,4}^{(2)}\pm d_{3,4}^{(2)}+d_{4,4}^{(2)}\right),$$

ⓘ

Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

►

22.9.18 $${\left(d_{1,4}^{(2)}\right)}^2{d}_{3,4}^{(2)}\pm {\left(d_{2,4}^{(2)}\right)}^2{d}_{4,4}^{(2)}+{\left(d_{3,4}^{(2)}\right)}^2{d}_{1,4}^{(2)}\pm {\left(d_{4,4}^{(2)}\right)}^2{d}_{2,4}^{(2)}=k^{\prime }\left(d_{1,4}^{(2)}\pm d_{2,4}^{(2)}+d_{3,4}^{(2)}\pm d_{4,4}^{(2)}\right),$$

ⓘ

Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

…

29: Bibliography H

… ►

Certain sums that contain cylindrical functions. Bul. Akad. Štiince RSS Moldoven. 1972 (3), pp. 75–77, 94 (Russian).

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External Links:

MathReview (D. P. Gupta)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

Encodings:

BibTeX

… ►

Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

ⓘ

External Links:

Document

Referenced by:

§15.18.

Encodings:

BibTeX

… ►

Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.

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Notes:

Double-precision Fortran, maximum accuracy 12D.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

… ►

Einige Eigenschaften der Dirichletschen Functionen $F(s)=\sum (\frac{D}{n})\cdot \frac{1}{n}$, die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten. Zeitschrift für Math. u. Physik 27, pp. 86–101 (German).

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External Links:

ZentralBlatt

Referenced by:

§25.11(i).

Encodings:

BibTeX

…

30: Bibliography B

… ►

Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 5, 109–117

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.77(iv).

Encodings:

BibTeX

… ►

A Fortran-90 based multiprecision system. ACM Trans. Math. Software 21 (4), pp. 379–387.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

… ►

Nouvelles Tables d’Intégrales Définies. P. Engels, Leide.

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Notes:

Corrected edition printed by Hafner Press, 1965.

Referenced by:

§1.4(iv).

Encodings:

BibTeX

►

Nouvelles Tables d’Intégrales Définies. G.E. Stechert & Co., New York.

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Referenced by:

§4.10(iii), §4.26(v), §4.40(v), §6.14(iii), §6.7(iv).

Encodings:

BibTeX

… ►

Angular Momentum. 3rd edition, Oxford University Press, Oxford.

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External Links:

ISBN 0-19-851759-9

Referenced by:

§34.9.

Encodings:

BibTeX

…