DLMF: Untitled Document

… ►

F. W. Schäfke and D. Schmidt (1966) Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III. Numer. Math. 8 (1), pp. 68–71.

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Notes:: Includes Algol program
External Links:: ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §28.36(ii).
Encodings:: BibTeX

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J. B. Seaborn (1991) Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

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External Links:: ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: Erratum (V1.2.0) Section 18.39.
Encodings:: BibTeX

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R. Shail (1978) Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.

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External Links:: ISSN 0020-7225, Document, MathReview, ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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A. Sidi (2004) Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective. Numer. Math. 98 (2), pp. 371–387.

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External Links:: ISSN 0029-599X, Document, MathReview (Iulian Coroian), ZentralBlatt
Referenced by:: §2.10(i).
Encodings:: BibTeX

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R. Spigler (1984) The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.

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External Links:: ISSN 0022-247X, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §1.13(v).
Encodings:: BibTeX

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A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:: ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.36(ii).
Encodings:: BibTeX

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M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:: ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(v).
Encodings:: BibTeX

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M. E. H. Ismail (2000a) An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.

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External Links:: ISSN 0030-8730, FullText, MathReview (T. Erber), ZentralBlatt
Referenced by:: §18.39(v).
Encodings:: BibTeX

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M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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External Links:: ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt
Referenced by:: §1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

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M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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Notes:: With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.
External Links:: ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt
Referenced by:: §18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

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… ►The Anger function

\mathbf{J}_{\nu}\left(z\right)

and Weber function

\mathbf{E}_{\nu}\left(z\right)

are defined by … ►

See accompanying text — Figure 11.10.1: Anger function $\mathbf{J}_{\nu}\left(x\right)$ for $-8\leq x\leq 8$ and $\nu=0,\frac{1}{2},1,\frac{3}{2}$ . Magnify

►

… ►

\mathbf{E}_{\nu}\left(-z\right)=-\mathbf{E}_{-\nu}\left(z\right).

… ►For

n=1,2,3,\dots

, …

§24.11 Asymptotic Approximations

… ►

24.11.1

(-1)^{n+1}B_{2n}\sim\frac{2(2n)!}{(2\pi)^{2n}},

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

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24.11.3

(-1)^{n}E_{2n}\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.4

(-1)^{n}E_{2n}\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

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24.11.6

(-1)^{\left\lfloor(n+1)/2\right\rfloor}\frac{\pi^{n+1}}{4(n!)}E_{n}\left(x% \right)\to\begin{cases}\sin\left(\pi x\right),&n\text{ even},\\ \cos\left(\pi x\right),&n\text{ odd},\end{cases}

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Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\sin\NVar{z}$ : sine function, $n$ : integer and $x$ : real or complex
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

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W. N. Bailey (1938) 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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M. V. Berry (1975) 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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L. C. Biedenharn and J. D. Louck (1981) 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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R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with

\exp(-x^{4})

. J. Approx. Theory 98, pp. 146–166.

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External Links:: ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

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W. Börsch-Supan (1960) 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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… ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by

K(\alpha)

,

E(\alpha)

,

\Pi(n\backslash\alpha)

,

F(\phi\backslash\alpha)

,

E(\phi\backslash\alpha)

, and

\Pi(n;\phi\backslash\alpha)

, where

\alpha=\operatorname{arcsin}k

and

n

is the

\alpha^{2}

(not related to

k

) in (19.1.1) and (19.1.2). Also, frequently in this reference

\alpha

is replaced by

m

and

\mathord{\backslash}\alpha

by

\mathord{|}m

, where

m=k^{2}

. However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. … ►

R_{F}\left(x,y,z\right)

,

R_{G}\left(x,y,z\right)

, and

R_{J}\left(x,y,z,p\right)

are the symmetric (in

x

,

y

, and

z

) integrals of the first, second, and third kinds; they are complete if exactly one of

x

,

y

, and

z

is identically 0. ►

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

is a multivariate hypergeometric function that includes all the functions in (19.1.3). …

… ►

•

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\,\mathrm{d}t$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$ . This takes the form $F(p,x)=4x/h(p,x)$ , approximately, where $h(p,x)=3(x^{2}-p)+\sqrt{(x^{2}-p)^{2}+8(x^{2}+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to\infty$ .

… ►

•

Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$ .

►

•

Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_{1}\left(z\right)$ and $z^{-1}\int_{0}^{z}t^{-1}(1-e^{-t})\,\mathrm{d}t$ for complex $z$ with $\left|\operatorname{ph}z\right|\leq\pi$ .

►

•

Verbeeck (1970) gives polynomial and rational approximations for $E_{p}\left(x\right)=(e^{-x}/x)P(z)$ , approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$ .

… ►

D. Naylor (1996) On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

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External Links:: ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

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G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

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G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.16), (11.11.17), §11.11(iii).
Encodings:: BibTeX

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E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.

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External Links:: ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §19.24(ii), §19.24(i), §19.9(i).
Encodings:: BibTeX

… ►

V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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… ►Further inequalities for

K\left(k\right)

and

E\left(k\right)

can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►Sharper inequalities for

F\left(\phi,k\right)

are: …(19.9.15) is useful when

k^{2}

and

{\sin}^{2}\phi

are both close to

1

, since the bounds are then nearly equal; otherwise (19.9.14) is preferable. ►Inequalities for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). … ►Other inequalities for

F\left(\phi,k\right)

can be obtained from inequalities for

R_{F}\left(x,y,z\right)

given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

… ►

E. Hairer, S. P. Nørsett, and G. Wanner (1993) Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

ⓘ

Notes:: A reprinting with corrections appeared in 2000.
External Links:: ISBN 3-540-56670-8, MathReview, ZentralBlatt
Referenced by:: §3.7(v), §3.7(i).
Encodings:: BibTeX

… ►

B. A. Hargrave and B. D. Sleeman (1977) 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

… ►

F. E. Harris (2000) Spherical Bessel expansions of sine, cosine, and exponential integrals. Appl. Numer. Math. 34 (1), pp. 95–98.

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External Links:: ISSN 0168-9274, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii), §6.10(ii), §6.15.
Encodings:: BibTeX

… ►

L. E. Hoisington and G. Breit (1938) Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

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External Links:: Document
Referenced by:: §33.7.
Encodings:: BibTeX

… ►

K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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External Links:: ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

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Cash App Wallet Phone%E2%98%8E%EF%B8%8F%2B1%28888%E2%80%92481%E2%80%924477%29%E2%98%8E%EF%B8%8F %22Number%22

21—30 of 859 matching pages

21: Bibliography S

22: Bibliography I

23: 11.10 Anger–Weber Functions

24: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

25: Bibliography B

26: 19.1 Special Notation

27: 8.27 Approximations

28: Bibliography N

29: 19.9 Inequalities

30: Bibliography H