DLMF: Untitled Document

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I. G. Macdonald (1990) Hypergeometric Functions.

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Notes:: Unpublished Lecture Notes, University of London. An online version exists.
External Links:: FullText
Referenced by:: §35.7(ii), §35.8(iii).
Encodings:: BibTeX

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H. Majima, K. Matsumoto, and N. Takayama (2000) Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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External Links:: ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt
Referenced by:: §13.12.
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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S. C. Milne (1985d) A

q

-analog of hypergeometric series well-poised in

\mathit{SU}(n)

and invariant

G

-functions. Adv. in Math. 58 (1), pp. 1–60.

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External Links:: ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §17.15.
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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A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:: ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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F. W. J. Olver (1991b) Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22 (5), pp. 1475–1489.

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External Links:: ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt
Referenced by:: §10.17(v), §10.40(iv), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).
Encodings:: BibTeX

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F. W. J. Olver (1993a) Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

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External Links:: ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt
Referenced by:: §10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).
Encodings:: BibTeX

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§6.20(i) Approximations in Terms of Elementary Functions

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$ -function (§13.2(i)) from which Chebyshev expansions near infinity for $E_{1}\left(z\right)$ , $\mathrm{f}\left(z\right)$ , and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

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J. Abad and J. Sesma (1995) Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.

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External Links:: ISSN 1047-5974, FullText
Referenced by:: §13.32(iii).
Encodings:: BibTeX

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D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

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G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992a) Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992b) Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.

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External Links:: MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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G. E. Andrews (1974) Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §17.16, Ch.17.
Encodings:: BibTeX

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… ►Here

\operatorname{erfc}

is the complementary error function (§7.2(i)), and …Also, … ►Expansions similar to (2.11.15) can be constructed for many other special functions. … … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

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A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

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W. G. Bickley and J. Nayler (1935) A short table of the functions

\mathrm{Ki}_{n}(x)

, from

n=1

to

n=16

. Phil. Mag. Series 7 20, pp. 343–347.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.43(iii), 2nd item.
Encodings:: BibTeX

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S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

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W. Bühring (1992) Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.

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External Links:: ISSN 0002-9939, FullText, MathReview (R. K. Raina), ZentralBlatt
Referenced by:: §16.4(ii), §16.8(ii).
Encodings:: BibTeX

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Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

A note about the multivalued nature of the Kummer confluent hypergeometric function of the second kind $U$ on the right-hand side of (7.18.10) was inserted.

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Paragraph Confluent Hypergeometric Functions (in §10.16)

Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

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Subsection 13.29(v)

A new Subsection Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

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Subsection 15.19(v)

A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:: ZentralBlatt
Referenced by:: §15.8(v), §15.8(v).
Encodings:: BibTeX

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G. Wei and B. E. Eichinger (1993) Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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External Links:: ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

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J. Wimp (1965) On the zeros of a confluent hypergeometric function. Proc. Amer. Math. Soc. 16 (2), pp. 281–283.

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External Links:: FullText, Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §13.9(ii).
Encodings:: BibTeX

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J. Wimp (1968) Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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External Links:: FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt
Referenced by:: §16.3(ii).
Encodings:: BibTeX

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L. Gatteschi (1990) New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §13.9(i), §13.9(ii).
Encodings:: BibTeX

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W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:: ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §13.29(iii), §15.19(iii).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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K. I. Gross and D. St. P. Richards (1987) Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.

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External Links:: ISSN 0002-9947, FullText, MathReview (Wayne J. Holman, III), ZentralBlatt
Referenced by:: §35.8(i), §35.8(ii), §35.8(iv), Ch.35.
Encodings:: BibTeX

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K. I. Gross and D. St. P. Richards (1991) Hypergeometric functions on complex matrix space. Bull. Amer. Math. Soc. (N.S.) 24 (2), pp. 349–355.

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External Links:: ISSN 0273-0979, Document, MathReview, ZentralBlatt
Referenced by:: §35.7(iii), §35.8(i), §35.8(ii), §35.8(iii), §35.8(iv).
Encodings:: BibTeX

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E. Pairman (1919) Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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External Links:: ZentralBlatt
Referenced by:: §5.1, Notations F.
Encodings:: BibTeX

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R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

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R. B. Paris (2013) Exponentially small expansions of the confluent hypergeometric functions. Appl. Math. Sci. (Ruse) 7 (133-136), pp. 6601–6609.

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External Links:: ISSN 1312-885X, Document, MathReview Entry
Referenced by:: §13.7(iii), §13.7(iii).
Encodings:: BibTeX

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W. F. Perger, A. Bhalla, and M. Nardin (1993) A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:: Extended-precision Fortran, maximum accuracy 12D.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

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hypergeometric%20functions

11—20 of 24 matching pages

11: Bibliography M

12: Bibliography O

13: 6.20 Approximations

§6.20(i) Approximations in Terms of Elementary Functions

14: Bibliography

15: 2.11 Remainder Terms; Stokes Phenomenon

16: Bibliography B

17: Errata

18: Bibliography W

19: Bibliography G

20: Bibliography P