DLMF: Untitled Document

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M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.12(i).
Encodings:: BibTeX

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G. E. Andrews and D. Foata (1980) Congruences for the

q

-secant numbers. European J. Combin. 1 (4), pp. 283–287.

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External Links:: ISSN 0195-6698, MathReview (J. E. Cigler), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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G. E. Andrews, I. P. Goulden, and D. M. Jackson (1986) Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

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External Links:: ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

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G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

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R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

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External Links:: FullText, MathReview (F. Schipp), ZentralBlatt
Referenced by:: §16.23(iii), (18.14.25), (18.14.26), (18.14.27), (18.38.3), Profile Richard A. Askey.
Encodings:: BibTeX

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G. Gasper and M. Rahman (1990) Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

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Notes:: A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).
External Links:: ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: (17.9.3), Erratum (V1.0.23) for Equation (17.9.3).
Encodings:: BibTeX

►

G. Gasper and M. Rahman (2004) Basic Hypergeometric Series. Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.

ⓘ

Notes:: With a foreword by Richard Askey
External Links:: ISBN 0-521-83357-4, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §17.1, §17.1, §17.10, (17.11.1), (17.11.2), (17.11.3), §17.13, §17.16, (17.4.5), (17.4.6), (17.4.7), (17.4.8), §17.4(i), §17.5, §17.6(i), §17.6(ii), §17.6(iii), §17.6(iv), §17.6(v), §17.7(i), §17.7(ii), §17.7(iii), §17.8, (17.9.3), §17.9(i), §17.9(ii), §17.9(iii), Ch.17, §18.27(i), §18.27(i), §18.27(ii), §18.27(iii), §18.27(iv), (18.28.24), (18.28.25), §18.28(i), §18.28(ii), §18.28(v), §18.28(viii), §26.19, Erratum (V1.0.10) for Section 17.1, Erratum (V1.0.23) for Equation (17.9.3).
Encodings:: BibTeX

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W. Gautschi (1964b) Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

ⓘ

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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W. Gautschi (1975) Computational Methods in Special Functions – A Survey. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35.

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External Links:: MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §19.36(ii), §19.5.
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2004c) Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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External Links:: ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §12.18.
Encodings:: BibTeX

…

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P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

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External Links:: MathReview (C. Tanaka)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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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

…

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10.32.14

K_{\nu}\left(z\right)=\frac{1}{2\pi^{2}i}\left(\frac{\pi}{2z}\right)^{\frac{1}% {2}}e^{-z}\cos\left(\nu\pi\right)\*\int_{-i\infty}^{i\infty}\Gamma\left(t% \right)\Gamma\left(\tfrac{1}{2}-t-\nu\right)\Gamma\left(\tfrac{1}{2}-t+\nu% \right)(2z)^{t}\,\mathrm{d}t,

\nu-\tfrac{1}{2}\notin\mathbb{Z},|\operatorname{ph}z|<\tfrac{3}{2}\pi

.

… ►

10.32.19

K_{\mu}\left(z\right)K_{\nu}\left(z\right)=\frac{1}{8\pi i}\int_{c-i\infty}^{c% +i\infty}\frac{\Gamma\left(t+\frac{1}{2}\mu+\frac{1}{2}\nu\right)\Gamma\left(t% +\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(t-\frac{1}{2}\mu+\frac{1}{2}% \nu\right)\Gamma\left(t-\frac{1}{2}\mu-\frac{1}{2}\nu\right)}{\Gamma\left(2t% \right)}(\tfrac{1}{2}z)^{-2t}\,\mathrm{d}t,

c>\tfrac{1}{2}(|\Re\mu|+|\Re\nu|),|\operatorname{ph}z|<\tfrac{1}{2}\pi

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter
Keywords:: Mellin transform
Referenced by:: §10.32(iii)
Permalink:: http://dlmf.nist.gov/10.32.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §10.32(iii), §10.32(iii), §10.32 and Ch.10

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B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

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Referenced by:: §18.38(i).
Encodings:: BibTeX

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M. J. Seaton (1982) Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: §33.1, §33.14(iv), 9th item.
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, M. Vianello, and F. Locatelli (1999) Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.

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External Links:: ISSN 0021-9045, Document, MathReview (Dumitru Acu), ZentralBlatt
Referenced by:: §2.9(iii).
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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T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

ⓘ

External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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R. Metzler, J. Klafter, and J. Jortner (1999) 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.

ⓘ

External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

A. E. Milne, P. A. Clarkson, and A. P. Bassom (1997) Bäcklund transformations and solution hierarchies for the third Painlevé equation. Stud. Appl. Math. 98 (2), pp. 139–194.

ⓘ

External Links:: ISSN 0022-2526, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.10(iii), §32.7(iii), §32.8(iii), §32.9(i).
Encodings:: BibTeX

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E. W. Montroll (1964) Lattice Statistics. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), University of California Engineering and Physical Sciences Extension Series, pp. 96–143.

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Notes:: Reprinted by Robert E. Krieger Publishing Co. in 1981.
External Links:: ZentralBlatt
Referenced by:: §26.20.
Encodings:: BibTeX

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

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

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S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

ⓘ

External Links:: FullText, MathReview (T. Oda), ZentralBlatt
Referenced by:: §21.6(ii).
Encodings:: BibTeX

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

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

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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

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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

…

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28.2.24

b_{n}\left(0\right)=n^{2},

n=1,2,3,\dots

.

ⓘ

Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $n$ : integer
Permalink:: http://dlmf.nist.gov/28.2.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(v), §28.2 and Ch.28

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See accompanying text — Figure 28.2.1: Eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ of Mathieu’s equation as functions of $q$ for $0\leq q\leq 10$ , $n=0,1,2,3,4$ ( $a$ ’s), $n=1,2,3,4$ ( $b$ ’s). Magnify

… ►

28.2.25

\begin{array}[]{rl}\mbox{for $q>0$:}&a_{0}<b_{1}<a_{1}<b_{2}<a_{2}<b_{3}<% \cdots,\\ \mbox{for $q<0$:}&a_{0}<a_{1}<b_{1}<b_{2}<a_{2}<a_{3}<\cdots.\end{array}

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.2.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(v), §28.2(v), §28.2 and Ch.28

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\operatorname{se}_{n}\left(z,0\right)=\sin\left(nz\right)

,

n=1,2,3,\dots

.

…

… ►

W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

ⓘ

External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

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A. P. Bassom, P. A. Clarkson, and A. C. Hicks (1995) Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.10(iv), §32.2(iii), §32.7(iv), §32.8(iv).
Encodings:: BibTeX

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R. Bo and R. Wong (1996) Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt
Referenced by:: §18.35(iii).
Encodings:: BibTeX

►

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.

ⓘ

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

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J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, and M. A. Shokrollahi (2001) Irregular primes and cyclotomic invariants to 12 million. J. Symbolic Comput. 31 (1-2), pp. 89–96.

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Notes:: Computational algebra and number theory (Milwaukee, WI, 1996)
External Links:: ISSN 0747-7171, Document, MathReview (Xavier-François Roblot), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

…

… ►If

f(z)

is continuous within and on a simple closed contour

C

and analytic within

C

, then … ►If

f(z)

is continuous within and on a simple closed contour

C

and analytic within

C

, and if

z_{0}

is a point within

C

, then … ►If

C

is a closed contour, and

z_{0}\not\in C

, then … ►Suppose

f(z)

is analytic in a domain

D

and

C_{1},C_{2}

are two arcs in

D

passing through

z_{0}

. … ►for any finite contour

C

in

D

. …

.%E9%98%BF%E6%A0%B9%E5%BB%B7%E4%B8%96%E7%95%8C%E6%9D%AF%E9%A2%84%E9%80%89%E8%B5%9B_%E3%80%8E%E7%BD%91%E5%9D%80%3A68707.vip%E3%80%8F%E4%B8%96%E7%95%8C%E6%9D%AF%E5%A1%9E%E5%B0%94%E7%BB%B4%E4%BA%9A%E5%AF%B9%E5%93%A5_b5p6v3_2022%E5%B9%B411%E6%9C%8828%E6%97%A521%E6%97%B621%E5%88%8647%E7%A7%92_6smwawga4

11—20 of 738 matching pages

11: Bibliography

12: Bibliography G

13: Bibliography H

14: 10.32 Integral Representations

15: Bibliography S

16: Bibliography M

17: Bibliography K

18: 28.2 Definitions and Basic Properties

19: Bibliography B

20: 1.9 Calculus of a Complex Variable