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11: 25.16 Mathematical Applications

… ►

25.16.2

\psi\left(x\right)=x-\frac{\zeta'\left(0\right)}{\zeta\left(0\right)}-\sum_{% \rho}\frac{x^{\rho}}{\rho}+o\left(1\right),

x\to\infty

ⓘ

Symbols:: $\psi\left(\NVar{x}\right)$ : Chebyshev $\psi$ -function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $o\left(\NVar{x}\right)$ : order less than and $x$ : real variable
Keywords:: asymptotic approximation
Source:: Apostol (2000, p. 9)
Permalink:: http://dlmf.nist.gov/25.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(i), §25.16 and Ch.25

… ►

25.16.5

H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},

ⓘ

Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer, $s$ : complex variable and $h(n)$ : harmonic number
Keywords:: Euler sum, infinite series, series representation
Source:: Apostol and Vu (1984, p. 85)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E5
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

►where

H_{n}

is given by (25.11.33). … ►

25.16.9

H\left(a\right)=\frac{a+2}{2}\zeta\left(a+1\right)-\frac{1}{2}\sum_{r=1}^{a-2}% \zeta\left(r+1\right)\zeta\left(a-r\right),

a=2,3,4,\dots

ⓘ

Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and $a$ : real or complex parameter
Source:: Apostol and Vu (1984, (12), p. 92)
Referenced by:: (25.16.8)
Permalink:: http://dlmf.nist.gov/25.16.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

… ►

H\left(s\right)

has a simple pole with residue

\zeta\left(1-2r\right)

(

=-B_{2r}/(2r)

) at each odd negative integer

s=1-2r

r=1,2,3,\dots

. …

12: Bibliography V

… ►

G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

… ►

A. L. Van Buren and J. E. Boisvert (2002) Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.

ⓘ

Notes:: For software, see Van Buren (website)
External Links:: ISSN 0033-569X, MathReview (Alan M. Cohen), ZentralBlatt
Referenced by:: §30.16(iii).
Encodings:: BibTeX

… ►

J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

ⓘ

Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

… ►

A. Verma and V. K. Jain (1983) Certain summation formulae for

q

-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

ⓘ

External Links:: ISSN 0019-5839, MathReview Entry, ZentralBlatt
Referenced by:: (17.6.4_5), §17.6(i), (17.8.8), §17.8.
Encodings:: BibTeX

… ►

M. N. Vrahatis, O. Ragos, T. Skiniotis, F. A. Zafiropoulos, and T. N. Grapsa (1995) RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions. Comput. Phys. Comm. 92 (2-3), pp. 252–266.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 16S. For corrections see same journal 117 (1999), p. 290
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

…

13: Bibliography W

… ►

E. Wagner (1986) Asymptotische Darstellungen der hypergeometrischen Funktion für große Parameter unterschiedlicher Größenordnung. Z. Anal. Anwendungen 5 (3), pp. 265–276 (German).

ⓘ

External Links:: ISSN 0232-2064, MathReview (R. P. Soni), ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

… ►

X. Wang and A. K. Rathie (2013) Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.

ⓘ

External Links:: ISSN 1687-1847, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §16.6, §16.6.
Encodings:: BibTeX

… ►

Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

ⓘ

External Links:: ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt
Referenced by:: §18.29.
Encodings:: BibTeX

… ►

R. Wong and J.-M. Zhang (1997) Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §18.34(iii).
Encodings:: BibTeX

… ►

P. Wynn (1966) Upon systems of recursions which obtain among the quotients of the Padé table. Numer. Math. 8 (3), pp. 264–269.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (F. L. Bauer), ZentralBlatt
Referenced by:: §3.11(iv).
Encodings:: BibTeX

14: 19.11 Addition Theorems

… ►

19.11.1

F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right),

ⓘ

Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle
Referenced by:: §19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

►

19.11.2

E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin% \theta\sin\phi\sin\psi.

ⓘ

Symbols:: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle
Permalink:: http://dlmf.nist.gov/19.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.5

\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left% (\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.6_5

R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right).

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos\NVar{z}$ : cosine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)
Permalink:: http://dlmf.nist.gov/19.11.E6_5
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.28):: This equation was given a number.
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.7

F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right),

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

…

15: 11.9 Lommel Functions

… ►Provided that

\mu\pm\nu\neq-1,-3,-5,\dots

, (11.9.1) has the general solution …where

A

B

are arbitrary constants,

s_{{\mu},{\nu}}\left(z\right)

is the Lommel function defined by … ►When

\mu\pm\nu\neq-1,-2,-3,\dots

, … ►For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). … ►For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2000, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32). …

16: 1.12 Continued Fractions

… ►

C_{n}

is called the

n

th approximant or convergent to $C$ .

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►when

p_{k}\not=0

k=1,2,3,\dots

. …when

c_{k}\not=0

k=1,2,3,\dots

. … ►Define …

17: Bibliography K

… ►

T. A. Kaeding (1995) Pascal program for generating tables of

\mathrm{SU}(3)

Clebsch-Gordan coefficients. Comput. Phys. Comm. 85 (1), pp. 82–88.

ⓘ

External Links:: ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

N. D. Kazarinoff (1988) Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

ⓘ

External Links:: ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt
Referenced by:: §16.23(iii).
Encodings:: BibTeX

… ►

T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

ⓘ

External Links:: Document, MathReview (M. Mikolas), ZentralBlatt
Referenced by:: (18.14.8), §18.14(i), §18.17(ii), §18.18(ii).
Encodings:: BibTeX

►

T. H. Koornwinder (1984a) Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups. In Special Functions: Group Theoretical Aspects and Applications, pp. 1–85.

ⓘ

Notes:: Reprinted by Kluwer Academic Publishers, Boston, 2002.
External Links:: ISBN 90-277-1822-9; 1-4020-0319-6, MathReview (Jacques Faraut), ZentralBlatt
Referenced by:: §14.31(ii), §15.17(iii), §15.9(ii).
Encodings:: BibTeX

… ►

B. G. Korenev (2002) Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

ⓘ

Notes:: Translated from the Russian by E. V. Pankratiev
External Links:: ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt
Referenced by:: §10.73(i), §10.73(i), §10.73(i).
Encodings:: BibTeX

…

18: Bibliography T

… ►

J. G. Taylor (1982) Improved error bounds for the Liouville-Green (or WKB) approximation. J. Math. Anal. Appl. 85 (1), pp. 79–89.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (Klaus Deimling), ZentralBlatt
Referenced by:: §2.7(iii).
Encodings:: BibTeX

… ►

N. M. Temme (1994a) A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.

ⓘ

Notes:: Includes Pascal program
External Links:: FullText
Referenced by:: §5.24(ii), §7.25(ii), §8.28(ii).
Encodings:: BibTeX

… ►

N. M. Temme (1995b) Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

ⓘ

External Links:: ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt
Referenced by:: §24.11, §24.16(i).
Encodings:: BibTeX

… ►

J. Todd (1975) The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.

ⓘ

Notes:: Collection of articles honoring Alston S. Householder. For corrigendum see same journal v. 18 (1975), no. 8, p. 462
External Links:: ISSN 0001-0782, Document, MathReview (L. Fox), ZentralBlatt
Referenced by:: §19.20(i), §19.20(iv), §19.36(iii).
Encodings:: BibTeX

… ►

O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

ⓘ

External Links:: Document
Referenced by:: §31.17(ii).
Encodings:: BibTeX

…

19: Bibliography O

… ►

K. Okamoto (1981) On the

\tau

-function of the Painlevé equations. Phys. D 2 (3), pp. 525–535.

ⓘ

External Links:: ISSN 0167-2789, Document, MathReview (D. A. Lutz), ZentralBlatt
Referenced by:: §32.6(i), §32.6(ii).
Encodings:: BibTeX

… ►

A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

ⓘ

External Links:: ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1964a) Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.

ⓘ

External Links:: FullText, MathReview (J.C.P. Miller), ZentralBlatt
Referenced by:: §3.6(iii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

ⓘ

External Links:: Document, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: §2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).
Encodings:: BibTeX

►

F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

ⓘ

External Links:: Document, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

…

20: 26.12 Plane Partitions

… ►As an example, there are six plane partitions of 3: … ►We define the

r\times s\times t

box

B(r,s,t)

as …Then the number of plane partitions in

B(r,s,t)

is … ►The number of symmetric plane partitions in

B(r,r,t)

is … ►The number of cyclically symmetric plane partitions in

B(r,r,r)

is …