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11: 14.17 Integrals

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14.17.1 $$\int {\left(1-x^2\right)}^{-\mu /2}{𝖯}_{\nu }^{\mu }\left(x\right)dx=-{\left(1-x^2\right)}^{-(\mu -1)/2}{𝖯}_{\nu }^{\mu -1}\left(x\right).$$

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Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(i), §14.17 and Ch.14

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14.17.2 $$\int {\left(1-x^2\right)}^{\mu /2}{𝖯}_{\nu }^{\mu }\left(x\right)dx=\frac{{\left(1-x^2\right)}^{(\mu +1)/2}}{(\nu -\mu )(\nu +\mu +1)}{𝖯}_{\nu }^{\mu +1}\left(x\right),$$ $\mu \ne \nu$ or $-\nu -1$.

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Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(i), §14.17 and Ch.14

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14.17.3 $$\int x{𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)dx=\frac{1}{2\nu (\nu +1)}\left(({\mu }^2-(\nu +1)(\nu +x^2)){𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)+(\nu +1)(\nu -\mu +1)x({𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)+{𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right))-{(\nu -\mu +1)}^2{𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)\right),$$ $\nu \ne 0,-1$.

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(i), §14.17 and Ch.14

… ►In (14.17.1)–(14.17.4), $𝖯$ may be replaced by $𝖰$, and in (14.17.3) and (14.17.4), $𝖰$ may be replaced by $𝖯$. … ►For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).

12: 10.9 Integral Representations

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10.9.30 $$J_{\nu }^2\left(z\right)+Y_{\nu }^2\left(z\right)=\frac{8}{{\pi }^2}{\int }_0^{\mathrm{\infty }}\cosh \left(2\nu t\right)K_0\left(2z\sinh t\right)dt,$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.9.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

… ►For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).

13: Bibliography G

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Positive sums of the classical orthogonal polynomials. SIAM J. Math. Anal. 8 (3), pp. 423–447.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (W. A. Al-Salam)

Referenced by:

(18.14.25), (18.14.26), (18.14.27).

Encodings:

BibTeX

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Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

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Notes:

For certification see same journal 8(1965), pp. 105–106, for remark see ACM Trans. Math. Software, 1(1975), pp. 282–284. Includes Algol program, maximum accuracy: 11D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(iii).

Encodings:

BibTeX

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Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.

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Notes:

Variable-precision Algol. For remark see ACM Trans. Math. Software 3(2) (1977), p. 204–205

External Links:

ISSN 0001-0782, Document

Referenced by:

§14.34(ii).

Encodings:

BibTeX

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Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

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External Links:

Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§10.74(iv), §14.32, §3.10(iii), §3.6(iii).

Encodings:

BibTeX

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Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 19(1964), no. 4, pp. 1–89

External Links:

Document, MathReview (A. Korányi), ZentralBlatt

Referenced by:

§35.3(i).

Encodings:

BibTeX

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14: 1.6 Vectors and Vector-Valued Functions

… ►Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over). … ►Note that $C$ can be given an orientation by means of $𝐜$. … ►with $(u,v)\in D$, an open set in the plane. … ►A surface is orientable if a continuously varying normal can be defined at all points of the surface. … ►If ${𝚽}_1$ and ${𝚽}_2$ are both orientation preserving or both orientation reversing parametrizations of $S$ defined on open sets $D_1$ and $D_2$ respectively, then …

15: Bibliography B

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:

ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

Referenced by:

§18.18(vii).

Encodings:

BibTeX

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Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces. J. Phys. A 8 (4), pp. 566–584.

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External Links:

Document, MathReview

Referenced by:

§36.14(iii).

Encodings:

BibTeX

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On the power function of the likelihood ratio test for MANOVA. J. Multivariate Anal. 82 (2), pp. 416–421.

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External Links:

ISSN 0047-259X, Document, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Angular Momentum in Quantum Physics: Theory and Application. Encyclopedia of Mathematics and its Applications, Vol. 8, Addison-Wesley Publishing Co., Reading, M.A..

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External Links:

ISBN 0-201-13507-8, MathReview (Kurt Bernardo Wolf), ZentralBlatt

Referenced by:

§34.14.

Encodings:

BibTeX

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Algorithm 21: Bessel function for a set of integer orders. Comm. ACM 3 (11), pp. 600.

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Notes:

Includes Algol code, minimum accuracy 5S. For certification see same journal 8(1965), p. 219.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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16: Bibliography M

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Inequalities for the zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 166–170.

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External Links:

ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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On the $G$-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.11(ii).

Encodings:

BibTeX

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.

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External Links:

Document, ZentralBlatt

Referenced by:

§28.7.

Encodings:

BibTeX

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17: 10.21 Zeros

… ►The parameter $t$ may be regarded as a continuous variable and ${\rho }_{\nu }$, ${\sigma }_{\nu }$ as functions ${\rho }_{\nu }(t)$, ${\sigma }_{\nu }(t)$ of $t$. … ►Let ${𝒞}_{\nu }\left(x\right)$, ${\rho }_{\nu }(t)$, and ${\sigma }_{\nu }(t)$ be defined as in §10.21(ii) and $M\left(x\right)$, $\theta \left(x\right)$, $N\left(x\right)$, and $\varphi \left(x\right)$ denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8. … ► $B_0(\zeta )$ and $C_0(\zeta )$ are defined by (10.20.11) and (10.20.12) with $k=0$. … ►Each curve that represents an infinite string of nonreal zeros should be located on the opposite side of its straight line asymptote. … ►Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions. …

18: Errata

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The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

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Version 1.0.11 (June 8, 2016)

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10.19.11 $$Q_3\left(a\right)=\frac{549}{28000}{a}^8-\frac{\mathrm{1\hspace{0.33em}10767}}{\mathrm{6\hspace{0.33em}93000}}{a}^5+\frac{79}{12375}{a}^2$$

Originally the first term on the right-hand side of this equation was written incorrectly as $-\frac{549}{28000}{a}^8$.

Reported 2015-03-16 by Svante Janson.

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Originally the Stirling number $s(10,6)$ was given incorrectly as 6327. The correct number is 63273.

$n$	$k$
	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
10	$0$	$-\mathrm{3\hspace{0.17em}62880}$	$\mathrm{10\hspace{0.17em}26576}$	$-\mathrm{11\hspace{0.17em}72700}$	$\mathrm{7\hspace{0.17em}23680}$	$-\mathrm{2\hspace{0.17em}69325}$	$63273$	$-9450$	$870$	$-45$	$1$

Reported 2013-11-25 by Svante Janson.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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19: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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20: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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$D_{-1}^{\pm }=0,$

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$D_0^{\pm }=1,$

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28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

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Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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