DLMF: Untitled Document

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I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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External Links:: ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

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P. Terwilliger (2013) The universal Askey-Wilson algebra and DAHA of type

(C^{\vee}_{1},C_{1})

. SIGMA 9, pp. Paper 047, 40 pp..

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External Links:: ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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I. Thompson (2013) Algorithm 926: incomplete gamma functions with negative arguments. ACM Trans. Math. Software 39 (2), pp. Art. 14, 9.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 7th item, §8.28(ii).
Encodings:: BibTeX

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W. J. Thompson (1994) Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:: With 1 Macintosh floppy disk (3.5 inch; DD). Programs for the calculation of $3j,6j,9j$ symbols are included.
External Links:: ISBN 0-471-55264-X, MathReview (J. F. Cornwell), ZentralBlatt
Referenced by:: §34.3(vii).
Encodings:: BibTeX

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A. A. Tuẑilin (1971) Theory of the Fresnel integral. USSR Comput. Math. and Math. Phys. 9 (4), pp. 271–279.

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Notes:: Translation of the paper in Zh. Vychisl. Mat. Mat. Fiz., Vol. 9, No. 4, 938-944, 1969
External Links:: Document, ZentralBlatt
Referenced by:: §7.13(iv).
Encodings:: BibTeX

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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, 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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T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:: MathReview (N. G. de Bruijn), ZentralBlatt
Referenced by:: (25.11.41), (25.11.42), §25.11(xii).
Encodings:: BibTeX

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H. Appel (1968) Numerical Tables for Angular Correlation Computations in

\alpha

-,

\beta

- and

\gamma

-Spectroscopy:

3j

-,

6j

-,

9j

-Symbols, F- and

\Gamma

-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:: ISBN 978-3-540-04218-1
Referenced by:: §34.14.
Encodings:: BibTeX

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F. M. Arscott (1959) A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

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External Links:: ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §29.11.
Encodings:: BibTeX

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M. Jimbo, T. Miwa, Y. Môri, and M. Sato (1980) Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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External Links:: ISSN 0167-2789, Document, MathReview, ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:: ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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N. Joshi and A. V. Kitaev (2005) The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:: ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt
Referenced by:: §32.11(i).
Encodings:: BibTeX

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N. Joshi and M. D. Kruskal (1992) The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.

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External Links:: ISSN 0022-2526, MathReview (I. P. Martynov), ZentralBlatt
Referenced by:: §32.11(i), §32.12(ii).
Encodings:: BibTeX

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G. Julia (1918) Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).

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External Links:: FullText, ZentralBlatt
Referenced by:: §3.8(viii).
Encodings:: BibTeX

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R. D. M. Garashchuk and J. C. Light (2001) Quasirandom distributed bases for bound problems. J. Chem. Phys. 114 (9), pp. 3929–3939.

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Referenced by:: §18.39(iii).
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

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

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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 (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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External Links:: ISSN 0001-0782, Document
Referenced by:: §33.26(ii).
Encodings:: BibTeX

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W. Gautschi (1967) Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

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External Links:: Document, MathReview (E. Frank), ZentralBlatt
Referenced by:: §10.74(iv), §14.32, §3.10(iii), §3.6(iii).
Encodings:: BibTeX

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J. Patera and P. Winternitz (1973) A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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External Links:: Document, ZentralBlatt
Referenced by:: §29.18(iv).
Encodings:: BibTeX

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M. D. Perlman and I. Olkin (1980) Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.

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External Links:: ISSN 0090-5364, FullText, MathReview (Takeaki Kariya), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

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E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

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M. J. D. Powell (1967) On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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External Links:: Document, MathReview, ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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Referenced by:: 2nd item.
Encodings:: BibTeX

…

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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). …

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9.9.6

a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.7

\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)% \right),

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $V$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.8

a^{\prime}_{k}=-U\left(\tfrac{3}{8}\pi(4k-3)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ , $k$ : nonnegative integer and $U$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.10

b_{k}=-T\left(\tfrac{3}{8}\pi(4k-3)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $b_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 21))
Permalink:: http://dlmf.nist.gov/9.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.21

W(t)\sim\pi^{-1/2}t^{-1/6}\left(1-\frac{7}{96}t^{-2}+\frac{1673}{6144}t^{-4}-% \frac{843\;94709}{265\;42080}t^{-6}+\frac{78\;02771\;35421}{1\;01921\;58720}t^% {-8}-\frac{20444\;90510\;51945}{6\;52298\;15808}t^{-10}+\cdots\right).

ⓘ

Defines:: $W$ : expansion (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion and $\pi$ : the ratio of the circumference of a circle to its diameter
Source:: Olver (1954, (A 19))
Permalink:: http://dlmf.nist.gov/9.9.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

…

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H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

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A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

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External Links:: ISSN 0001-0782, Document
Referenced by:: §5.24(ii).
Encodings:: BibTeX

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V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §9.1, Notations U, Notations V.
Encodings:: BibTeX

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C. K. Frederickson and P. L. Marston (1994) Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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External Links:: Document
Referenced by:: §36.14(iv).
Encodings:: BibTeX

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B. R. Frieden (1971) Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions. In Progress in Optics, E. Wolf (Ed.), Vol. 9, pp. 311–407.

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Referenced by:: §30.15(i), §30.15(ii), §30.15(iii), §30.15(iv), §30.15(v), §30.15(v).
Encodings:: BibTeX

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

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

…

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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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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.

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External Links:: FullText
Referenced by:: §8.24(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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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

…

.%E6%96%B0%E5%AE%9D2%E4%B8%96%E7%95%8C%E6%9D%AF%E7%BB%93%E6%9D%9F%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.tbx-k2q1w9-2022%E5%B9%B411%E6%9C%8829%E6%97%A54%E6%97%B659%E5%88%8617%E7%A7%92

11—20 of 626 matching pages

11: Bibliography T

12: Bibliography

13: Bibliography J

14: Bibliography G

15: Bibliography P

16: 34.5 Basic Properties: $\mathit{6j}$ Symbol

17: 9.9 Zeros

18: Bibliography F

19: Bibliography B

20: Bibliography M