.%E5%A4%AA%E9%98%B3%E5%9F%8E3%E5%A8%B1%E4%B9%90%E4%B8%96%E7%95%8C%E6%9D%AF%E5%B9%B3%E5%8F%B0%E4%BB%A3%E7%90%86%E3%80%8E%E4%B8%96%E7%95%8C%E6%9D%AF%E4%BD%A3%E9%87%91%E5%88%86%E7%BA%A255%25%EF%BC%8C%E5%92%A8%E8%AF%A2%E4%B8%93%E5%91%98%EF%BC%9A%40ky975%E3%80%8F.qaq-k2q1w9-2022%E5%B9%B411%E6%9C%8830%E6%97%A59%E6%97%B651%E5%88%8655%E7%A7%92

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11—20 of 631 matching pages

11: 34.5 Basic Properties: $\mathit{6j}$ Symbol

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34.5.11

{(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

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Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half., $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

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34.5.13

E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.

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Defines:: $E(j)$ (locally)
Symbols:: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

►For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99). …

12: 10.60 Sums

… ►For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). …

13: Bibliography H

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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

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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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J. R. Herndon (1961b) Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

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Notes:: Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

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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

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A. Hurwitz (1882) Einige Eigenschaften der Dirichletschen Functionen

F(s)=\sum(\frac{D}{n})\cdot\frac{1}{n}

, die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten. Zeitschrift für Math. u. Physik 27, pp. 86–101 (German).

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External Links:: ZentralBlatt
Referenced by:: §25.11(i).
Encodings:: BibTeX

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14: Bibliography K

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K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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External Links:: ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

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M. Kaneko (1997) Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

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External Links:: ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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M. K. Kerimov (2008) Overview of some new results concerning the theory and applications of the Rayleigh special function. Comput. Math. Math. Phys. 48 (9), pp. 1454–1507.

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External Links:: ISSN 0965-5425, Document, FullText
Referenced by:: §10.21(xiii), §10.21(xiii).
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.

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

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E. Kreyszig (1957) On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.

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External Links:: MathReview (W. C. Rheinboldt), ZentralBlatt
Referenced by:: §7.13(iii).
Encodings:: BibTeX

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15: 25.5 Integral Representations

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25.5.7

\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)% !}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x% ^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\,\mathrm{d}x,

\Re s>-(2n+1)

n=1,2,3,\dots

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: improper integral, integral representation
Proof sketch:: Derivable from (25.5.6) by adding and subtracting $\sum_{m=1}^{n}B_{2m}x^{2m-1}/(2m)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that $({\mathrm{e}}^{x}-1)^{-1}-x^{-1}+1/2-\sum_{m=1}^{n}B_{2m}x^{2m-1}/(2m)!=O\left% (x^{2n+1}\right)$ as $x\to 0$ , demonstrating the region of convergence.
Referenced by:: Erratum (V1.0.9) for Chapters 7, 25
Permalink:: http://dlmf.nist.gov/25.5.E7
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.9):: We have rewritten the original summation $\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}\frac{\Gamma\left(s+2m-1\right)}{\Gamma\left% (s\right)}$ more concisely as $\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}{\left(s\right)_{2m-1}}$ using the Pochhammer symbol.
See also:: Annotations for §25.5(i), §25.5 and Ch.25

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25.5.10

\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\left(% s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\left(\frac{1}{2}\pi x% \right)}\,\mathrm{d}x.

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $x$ : real variable and $s$ : complex variable
Keywords:: improper integral, integral representation
Sources:: Lindelöf (1905, (6), p. 98); with $x=2t$ ; Erdélyi et al. (1953a, (1.12.15), p. 33)
Permalink:: http://dlmf.nist.gov/25.5.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §25.5(i), §25.5 and Ch.25

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25.5.11

\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin% \left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\,\mathrm{% d}x.

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $x$ : real variable and $s$ : complex variable
Keywords:: improper integral, integral representation
Sources:: Lindelöf (1905, (3), p. 97); Erdélyi et al. (1953a, (1.12.12), p. 33)
Permalink:: http://dlmf.nist.gov/25.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §25.5(i), §25.5 and Ch.25

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25.5.12

\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin\left(% s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\,\mathrm{d}x.

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $x$ : real variable and $s$ : complex variable
Keywords:: improper integral, integral representation
Source:: Lindelöf (1905, (4), p. 98); with $x=2t$
Permalink:: http://dlmf.nist.gov/25.5.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §25.5(i), §25.5 and Ch.25

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16: Bibliography S

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H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:: FullText, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §3.5(viii).
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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K. Srinivasa Rao, V. Rajeswari, and C. B. Chiu (1989) A new Fortran program for the

9

j

angular momentum coefficient. Comput. Phys. Comm. 56 (2), pp. 231–248.

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External Links:: Document, ZentralBlatt
Referenced by:: §34.13.
Encodings:: BibTeX

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S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:: ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §17.3(i).
Encodings:: BibTeX

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17: Bibliography N

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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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N. Nielsen (1909) Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopoldina 90, pp. 123–212.

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External Links:: ZentralBlatt
Referenced by:: §25.12(i).
Encodings:: BibTeX

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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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Numerical Recipes (commercial C, C++, Fortran 77, and Fortran 90 libraries)

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Notes:: A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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18: Bibliography M

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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.

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

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J. McMahon (1894) On the roots of the Bessel and certain related functions. Ann. of Math. 9 (1-6), pp. 23–30.

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External Links:: ISSN 0003-486X, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

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J. Meixner (1934) Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. Lond. Math. Soc. 9, pp. 6–13 (German).

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External Links:: ISSN 0024-6107; 1469-7750/e, ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

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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.

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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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T. Morita (2013) A connection formula for the

q

-confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.

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External Links:: ISSN 1815-0659, Document, MathReview (Jeremy Lovejoy), ZentralBlatt
Referenced by:: §17.6(ii), §17.6(ii).
Encodings:: BibTeX

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19: Bibliography C

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R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:: ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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B. C. Carlson (1978) Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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External Links:: Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §19.26(i).
Encodings:: BibTeX

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T. S. Chihara and M. E. H. Ismail (1993) Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

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External Links:: ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt
Referenced by:: §18.28(iv).
Encodings:: BibTeX

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J. S. Christiansen and M. E. H. Ismail (2006) A moment problem and a family of integral evaluations. Trans. Amer. Math. Soc. 358 (9), pp. 4071–4097.

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External Links:: ISSN 0002-9947, Document, MathReview (Roelof Koekoek), ZentralBlatt
Referenced by:: §18.28(iv).
Encodings:: BibTeX

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W. J. Cody and K. E. Hillstrom (1967) Chebyshev approximations for the natural logarithm of the gamma function. Math. Comp. 21 (98), pp. 198–203.

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External Links:: FullText, ZentralBlatt
Referenced by:: §5.23(i).
Encodings:: BibTeX

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20: Bibliography B

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R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:: ISSN 0893-9659, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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K. Bay, W. Lay, and A. Akopyan (1997) Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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External Links:: ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt
Referenced by:: §31.17(ii).
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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T. H. Boyer (1969) Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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External Links:: Document, MathReview (C. V. Coffman), ZentralBlatt
Referenced by:: §10.21(xi).
Encodings:: BibTeX

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R. Bulirsch (1967) Numerical calculation of the sine, cosine and Fresnel integrals. Numer. Math. 9 (5), pp. 380–385.

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Notes:: Algol 60 procedures for sine, cosine, and 2 Fresnel integrals are included.
External Links:: ISSN 0029-599X, ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt
Referenced by:: 2nd item, §7.25(iv).
Encodings:: BibTeX

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