DLMF: Untitled Document

… ►

S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

ⓘ

External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

… ►

D. E. Amos (1983c) Uniform asymptotic expansions for exponential integrals

E_{n}(x)

and Bickley functions

{\rm{K}i}_{n}(x)

. ACM Trans. Math. Software 9 (4), pp. 467–479.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

K. Aomoto (1987) Special value of the hypergeometric function

{}_{3}F_{2}

and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

ⓘ

External Links:: ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

… ►

V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups

{A}_{k},{D}_{k},{E}_{k}

and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

ⓘ

Notes:: In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.
External Links:: MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

►

V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

ⓘ

Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

…

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of

\sum_{k=1}^{m}k^{n}

,

m=1(1)100

,

n=1(1)10

;

\sum_{k=1}^{\infty}k^{-n}

,

\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}

,

\sum_{k=0}^{\infty}(2k+1)^{-n}

,

n=1,2,\dotsc

, 20D;

\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}

,

n=1,2,\dotsc

, 18D. ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

… ►Beginning with

u_{n+1}=0

,

u_{n}=c_{n}

, we apply … ►With

b_{0}=1

, the last

q

equations give

b_{1},\dots,b_{q}

as the solution of a system of linear equations. … ►(3.11.29) is a system of

n+1

linear equations for the coefficients

a_{0},a_{1},\dots,a_{n}

. … ►With this choice of

a_{k}

and

f_{j}=f(x_{j})

, the corresponding sum (3.11.32) vanishes. … ►Two are endpoints:

(x_{0},y_{0})

and

(x_{3},y_{3})

; the other points

(x_{1},y_{1})

and

(x_{2},y_{2})

are control points. …

… ►

S. K. Kim (1972) The asymptotic expansion of a hypergeometric function

{}_{2}F_{2}(1,\,\alpha;\,\rho_{1},\,\rho_{2};\,z)

. Math. Comp. 26 (120), pp. 963.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §16.11(ii), §16.11(ii).
Encodings:: BibTeX

… ►

Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series

{}_{r+3}F_{r+2}

. Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

ⓘ

External Links:: ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt
Referenced by:: §16.4(ii), §16.4(ii).
Encodings:: BibTeX

… ►

C. G. Kokologiannaki, P. D. Siafarikas, and C. B. Kouris (1992) On the complex zeros of

H_{\mu}(z),\ J^{\prime}_{\mu}(z),\ J^{\prime\prime}_{\mu}(z)

for real or complex order. J. Comput. Appl. Math. 40 (3), pp. 337–344.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

… ►

T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

ⓘ

External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

… ►

E. D. Krupnikov and K. S. Kölbig (1997) Some special cases of the generalized hypergeometric function

{}_{q+1}F_{q}

. J. Comput. Appl. Math. 78 (1), pp. 79–95.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §16.7.
Encodings:: BibTeX

…

… ►

K. Okamoto (1986) Studies on the Painlevé equations. III. Second and fourth Painlevé equations,

P_{{\rm II}}

and

P_{\mathrm{IV}}

. Math. Ann. 275 (2), pp. 221–255.

ⓘ

External Links:: ISSN 0025-5831, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(ii), §32.10(iv), §32.6(iii), §32.6(v), §32.7(viii).
Encodings:: BibTeX

►

K. Okamoto (1987a) Studies on the Painlevé equations. I. Sixth Painlevé equation

P_{{\rm VI}}

. Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

ⓘ

External Links:: ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).
Encodings:: BibTeX

►

K. Okamoto (1987b) Studies on the Painlevé equations. II. Fifth Painlevé equation

P_{\rm V}

. Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

ⓘ

External Links:: ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(v), §32.6(v), §32.7(viii).
Encodings:: BibTeX

►

K. Okamoto (1987c) Studies on the Painlevé equations. IV. Third Painlevé equation

P_{{\rm III}}

. Funkcial. Ekvac. 30 (2-3), pp. 305–332.

ⓘ

External Links:: ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(iii), §32.2(iii), §32.6(iv), §32.7(viii).
Encodings:: BibTeX

… ►

H. Oser (1960) Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

ⓘ

Notes:: Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.
External Links:: ISSN 0001-0782, Document
Referenced by:: §10.77(ii).
Encodings:: BibTeX

…

… ►

Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

.

► ►►►

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
…
$11\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
…

► ►

4.17.1

\lim_{z\to 0}\frac{\sin z}{z}=1,

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.74
Permalink:: http://dlmf.nist.gov/4.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

►

4.17.2

\lim_{z\to 0}\frac{\tan z}{z}=1.

ⓘ

Symbols:: $\tan\NVar{z}$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.75
Permalink:: http://dlmf.nist.gov/4.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

►

4.17.3

\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

… ►Also let

\xi=2\sqrt{h}\cos x

and

D_{m}\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)

(§18.3). … ►

\sigma_{m}\sim 1+\dfrac{s}{2^{3}h}+\dfrac{4s^{2}+3}{2^{7}h^{2}}+\dfrac{19s^{3}% +59s}{2^{11}h^{3}}+\cdots,

… ►

28.8.11

P_{m}(x)\sim 1+\dfrac{s}{2^{3}h{\cos}^{2}x}+\dfrac{1}{h^{2}}\left(\dfrac{s^{4}% +86s^{2}+105}{2^{11}{\cos}^{4}x}-\dfrac{s^{4}+22s^{2}+57}{2^{11}{\cos}^{2}x}% \right)+\cdots,

ⓘ

Defines:: $P_{m}(x)$ (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cos\NVar{z}$ : cosine function, $m$ : integer, $h$ : parameter and $x$ : real variable
A&S Ref:: 20.9.14 (in different form)
Permalink:: http://dlmf.nist.gov/28.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(iii), §28.8 and Ch.28

… ►The approximations are expressed in terms of Whittaker functions

W_{\kappa,\mu}\left(z\right)

and

M_{\kappa,\mu}\left(z\right)

with

\mu=\tfrac{1}{4}

; compare §2.8(vi). … ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions

\operatorname{me}_{\nu}\left(z,q\right)

(§28.12(ii)) and modified Mathieu functions

{\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)

(§28.20(iii)). …

… ►

L. Habsieger (1986) La

q

-conjecture de Macdonald-Morris pour

G_{2}

. C. R. Acad. Sci. Paris Sér. I Math. 303 (6), pp. 211–213 (French).

ⓘ

External Links:: ISSN 0249-6291, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

… ►

R. S. Heller (1976) 25D Table of the First One Hundred Values of

j_{0,s}

,

J_{1}(j_{0,s})

,

j_{1,s}

,

J_{0}(j_{1,s})=J_{0}(j^{\prime}_{0,s+1})

,

j^{\prime}_{1,s}

,

J_{1}(j^{\prime}_{1,s})

. Technical report Department of Physics, Worcester Polytechnic Institute, Worcester, MA.

ⓘ

Notes:: Ms. of six pages deposited in the UMT file
Referenced by:: 8th item.
Encodings:: BibTeX

… ►

D. R. Herrick and S. O’Connor (1998) Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

ⓘ

External Links:: Document
Referenced by:: §15.18.
Encodings:: BibTeX

… ►

H. W. Hethcote (1970) Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(xiv).
Encodings:: BibTeX

… ►

G. W. Hill (1981) Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions:

I_{1}(x)/I_{0}(x)

,

I_{1.5}(x)/I_{0.5}(x)

and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

ⓘ

Notes:: Single-precision Fortran, minimum accuracy 8S.
External Links:: ISSN 0098-3500, Document, GAMS (module 571; package TOMS)
Referenced by:: 5th item.
Encodings:: BibTeX

…

… ►For the generalized hypergeometric function

{{}_{3}F_{2}}

see (16.2.1). … ►Define operators

K_{0}

and

K_{1}

acting on symmetric Laurent polynomials by

K_{0}=L

(

L

given by (18.28.6_2)) and

(K_{1}f)(z)=(z+z^{-1})f(z)

. …commutes with

K_{0},K_{1},K_{2}

, that is

K_{j}Q=QK_{j}

, and satisfies …where

Q_{0}

is a constant with explicit expression in terms of

e_{1},e_{2},e_{3},e_{4}

and

q

given in Koornwinder (2007a, (2.8)). ►The abstract associative algebra with generators

K_{0},K_{1},K_{2}

and relations (18.38.4), (18.38.6) and with the constants

B,C_{0},C_{1},D_{0},D_{1}

in (18.38.6) not yet specified, is called the Zhedanov algebra or Askey–Wilson algebra AW(3). …

… ►

20.9.1

k={\theta_{2}}^{2}\left(0\middle|\tau\right)/{\theta_{3}}^{2}\left(0\middle|% \tau\right)

ⓘ

Defines:: $k$ : modulus (locally)
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function and $\tau$ : lattice parameter
Referenced by:: §20.11(iii), §20.9(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/20.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

… ►

20.9.3

R_{F}\left(\frac{{\theta_{2}}^{2}\left(z,q\right)}{{\theta_{2}}^{2}\left(0,q% \right)},\frac{{\theta_{3}}^{2}\left(z,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)},\frac{{\theta_{4}}^{2}\left(z,q\right)}{{\theta_{4}}^{2}\left(0,q% \right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{\theta_{1}\left(z,q\right)}z,

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §19.25(v), §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

►

20.9.4

R_{F}\left(0,{\theta_{3}}^{4}\left(0,q\right),{\theta_{4}}^{4}\left(0,q\right)% \right)=\tfrac{1}{2}\pi,

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $q$ : nome
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

►

20.9.5

\exp\left(-\frac{\pi R_{F}\left(0,k^{2},1\right)}{R_{F}\left(0,{k^{\prime}}^{2% },1\right)}\right)=q.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $q$ : nome and $k$ : modulus
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

… ►See Koblitz (1993, Ch. 2, §4) and Titchmarsh (1986b, pp. 21–22). …

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21—30 of 789 matching pages

21: Bibliography

22: 24.20 Tables

23: 3.11 Approximation Techniques

24: Bibliography K

25: Bibliography O

26: 4.17 Special Values and Limits

27: 28.8 Asymptotic Expansions for Large $q$

28: Bibliography H

29: 18.38 Mathematical Applications

30: 20.9 Relations to Other Functions