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21: 9.7 Asymptotic Expansions

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9.7.1 $$\zeta =\frac{2}{3}{z}^{3/2}.$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$z$: complex variable

Permalink:

http://dlmf.nist.gov/9.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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$$u_k=\frac{(2k+1)(2k+3)(2k+5)\mathrm{\cdots }(6k-1)}{{216}^{k}k!}=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}{u}_{k-1},$$

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§9.7(ii) Poincaré-Type Expansions

… ►where $\xi =\frac{2}{3}{x}^{3/2}$. … ►

9.7.17

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable, $\zeta$: change of variable, $\chi (x)$: function, $n$: index and $\sigma$: real variable

Proof sketch:

Derivable from (9.6.1)–(9.6.5) and Nemes (2017b, (26) and Propositions B.1, B.3, B.4).

Referenced by:

§9.7(iv), Erratum (V1.0.11) for Equation (9.7.17)

Permalink:

http://dlmf.nist.gov/9.7.E17

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

The bounds have been sharpened for $|\mathrm{ph}z|\le \frac{1}{3}\pi$, from $2\exp \left({\displaystyle \frac{\sigma }{36|\zeta |}}\right)$ to $1$; for $\frac{1}{3}\pi \le |\mathrm{ph}z|\le \frac{2}{3}\pi$, from $2\chi (n)\exp \left({\displaystyle \frac{\sigma \pi }{72|\zeta |}}\right)$ to $\min (|\csc \left(\mathrm{ph}\zeta \right)|,\chi (n+\sigma )+1)$; and for from ${\displaystyle \frac{4\chi (n)}{{|\cos \left(\mathrm{ph}\zeta \right)|}^n}}\exp \left({\displaystyle \frac{\sigma \pi }{36|\mathrm{\Re }\zeta |}}\right)$ to $\frac{\sqrt{2\pi (n+\sigma )}}{{|\cos \left(\mathrm{ph}\zeta \right)|}^{n+\sigma }}+\chi (n+\sigma )+1$.

Errata (effective with 1.0.11):

Originally the constraint condition was written incorrectly as $\frac{2}{3}\pi \le |\mathrm{ph}z|\le \pi$. Also, the equation was reformatted to display the constraints in the equation instead of in the text.

Reported 2014-11-05 by Gergő Nemes

See also:

Annotations for §9.7(iv), §9.7 and Ch.9

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22: Bibliography H

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

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External Links:

ISSN 0002-9890, Document, Link, MathReview Entry

Referenced by:

§1.16(ix).

Encodings:

BibTeX

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On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.

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External Links:

ISSN 0033-5614, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(i), §32.17.

Encodings:

BibTeX

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Exponential sums and lattice points. III. Proc. London Math. Soc. (3) 87 (3), pp. 591–609.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (Angel V. Kumchev), ZentralBlatt

Referenced by:

§27.11, §27.11, Erratum (V1.1.8) for Section 27.11.

Encodings:

BibTeX

23: 19.25 Relations to Other Functions

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19.25.13 $$D(\varphi ,k)=\frac{1}{3}{R}_D(c-1,c-k^2,c).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.25.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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24: 20.8 Watson’s Expansions

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20.8.1 $$\frac{{\theta }_2(0,q){\theta }_3(z,q){\theta }_4(z,q)}{{\theta }_2(z,q)}=2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n^2}{\mathrm{e}}^{\mathrm{i}2nz}}{q^{-n}{\mathrm{e}}^{-\mathrm{i}z}+q^n{\mathrm{e}}^{\mathrm{i}z}}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.8 and Ch.20

… ►This reference and Bellman (1961, pp. 46–47) include other expansions of this type.

25: 15.9 Relations to Other Functions

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15.9.26 $$D\left(k\right)=\frac{\pi }{4}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{3}{2}}{2};k^2).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $D\left(k\right)$: complete elliptic integral of Legendre’s type and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E26

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

26: 23.10 Addition Theorems and Other Identities

… ►For further addition-type identities for the $\sigma$-function see Lawden (1989, §6.4). … ►(23.10.8) continues to hold when $e_1$, $e_2$, $e_3$ are permuted cyclically. … ►For $n=2,3,\mathrm{\dots }$, … ►

23.10.15 $$A_n={\left(\frac{{\pi }^2{G}^2}{{\omega }_1}\right)}^{n^2-1}\frac{q^{n(n-1)/2}}{{\mathrm{i}}^{n-1}}\exp \left(-\frac{(n-1){\eta }_1}{3{\omega }_1}\left((2n-1)({\omega }_1^2+{\omega }_3^2)+3(n-1){\omega }_1{\omega }_3\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $q$: nome, $n$: integer, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $A_n$: coefficients, $G$ and ${\eta }_j$: complex number

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iii), §23.10 and Ch.23

… ►Also, when $𝕃$ is replaced by $c𝕃$ the lattice invariants $g_2$ and $g_3$ are divided by $c^4$ and $c^6$, respectively. …

27: 19.5 Maclaurin and Related Expansions

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19.5.3 $$D\left(k\right)=\frac{\pi }{4}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{3}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{(m+1)!m!}{k}^{2m}=\frac{\pi }{4}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{3}{2},\frac{1}{2}}{2};k^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $D\left(k\right)$: complete elliptic integral of Legendre’s type, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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28: Bibliography G

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. J. Comput. Appl. Math. 153 (1-2), pp. 225–234.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.45, §10.74(viii).

Encodings:

BibTeX

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Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

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29: Bibliography K

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

Encodings:

BibTeX

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Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

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External Links:

ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

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External Links:

ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Askey-Wilson Polynomials for Root Systems of Type $BC$ . In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

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External Links:

FullText, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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30: 18.11 Relations to Other Functions

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18.11.3 $$H_n\left(x\right)=2^{n}U(-\frac{1}{2}n,\frac{1}{2},x^2)=2^{n}xU(-\frac{1}{2}n+\frac{1}{2},\frac{3}{2},x^2)=2^{\frac{1}{2}n}{\mathrm{e}}^{\frac{1}{2}{x}^2}U(-n-\frac{1}{2},2^{\frac{1}{2}}x),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.55, 22.5.58 ((factor $x$ missing on RHS of 22.5.55))

Referenced by:

§18.11(i), §18.15(v)

Permalink:

http://dlmf.nist.gov/18.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

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18.11.4 $${\mathrm{𝐻𝑒}}_n\left(x\right)=2^{\frac{1}{2}n}U(-\frac{1}{2}n,\frac{1}{2},\frac{1}{2}{x}^2)=2^{\frac{1}{2}(n-1)}xU(-\frac{1}{2}n+\frac{1}{2},\frac{3}{2},\frac{1}{2}{x}^2)={\mathrm{e}}^{\frac{1}{4}{x}^2}U(-n-\frac{1}{2},x).$$

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Symbols:

${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.59

Permalink:

http://dlmf.nist.gov/18.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

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§18.11(ii) Formulas of Mehler–Heine Type

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