DLMF: Untitled Document

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P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
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

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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

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G. H. Hardy and S. Ramanujan (1918) Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, pp. 75–115.

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Notes:: Reprinted in Ramanujan (1962, pp. 276–309).
External Links:: Document, MathReview Entry, ZentralBlatt
Referenced by:: §27.14(iii).
Encodings:: BibTeX

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G. H. Hardy (1940) Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press, Cambridge, England.

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Notes:: Reprinted by Chelsea Publishing Company, New York, 1959.
External Links:: MathReview (H. Rademacher), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

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External Links:: ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt
Referenced by:: §27.14(v).
Encodings:: BibTeX

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S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:: §24.7(i).
Encodings:: BibTeX

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S. Ramanujan (1962) Collected Papers of Srinivasa Ramanujan. Chelsea Publishing Co., New York.

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Notes:: Reprinted with commentary by Bruce Berndt, AMS Chelsea Publishing Series, American Mathematical Society (2000).
External Links:: ISBN 0-8218-2076-1
Referenced by:: §8.11(v), G. H. Hardy and S. Ramanujan (1918).
Encodings:: BibTeX

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P. Ribenboim (1979) 13 Lectures on Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:: ISBN 0-387-90432-8, MathReview (T. Metsänkylä), ZentralBlatt
Referenced by:: §24.10(iv), §24.17(iii).
Encodings:: BibTeX

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K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii).
Encodings:: BibTeX

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5.13.1

{\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a\right)\Gamma\left% (b-s\right)z^{-s}\,\mathrm{d}s=\frac{\Gamma\left(a+b\right)z^{a}}{(1+z)^{a+b}}},

\Re\left(a+b\right)>0

,

-\Re a<c<\Re b

,

|\operatorname{ph}z|<\pi

.

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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, $\operatorname{ph}$ : phase, $\Re$ : real part, $s$ : real or complex variable, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13, §5.13, §5.19(ii)
Permalink:: http://dlmf.nist.gov/5.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13 and Ch.5

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Ramanujan’s Beta Integral

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Ramanujan’s Integrals

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D. F. Lawden (1989) Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

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External Links:: ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.
Encodings:: BibTeX

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D. H. Lehmer (1943) Ramanujan’s function

\tau(n)

. Duke Math. J. 10 (3), pp. 483–492.

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External Links:: Document, MathReview (H. S. Zuckerman), ZentralBlatt
Referenced by:: §27.20, §27.20, §27.21.
Encodings:: BibTeX

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D. H. Lehmer (1947) The vanishing of Ramanujan’s function

\tau(n)

. Duke Math. J. 14 (2), pp. 429–433.

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External Links:: Document, MathReview (R. A. Rankin), ZentralBlatt
Referenced by:: §27.14(vi).
Encodings:: BibTeX

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D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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J. Lepowsky and R. L. Wilson (1982) A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.16.
Encodings:: BibTeX

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… ► Schempp), published by Reidel Publishing Company in 1984, Ramanujan Revisited (with G. … ►S. … ►Askey was a member of the original editorial committee for the DLMF project, serving as an Associate Editor advising on all aspects of the project from the mid-1990’s to the mid-2010’s when the organizational structure of the DLMF project was reconstituted; see About the Project.

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A. J. E. M. Janssen (2021) Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:: ISSN 1389-2355
External Links:: FullText
Referenced by:: §7.8, §7.8.
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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F. Johansson (2012) Efficient implementation of the Hardy-Ramanujan-Rademacher formula. LMS J. Comput. Math. 15, pp. 341–359.

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External Links:: ISSN 1461-1570, Document, Link, MathReview (Tim Huber), ZentralBlatt
Referenced by:: §27.20.
Encodings:: BibTeX

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D. S. Jones (1972) Asymptotic behavior of integrals. SIAM Rev. 14 (2), pp. 286–317.

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External Links:: Document, MathReview (A. Erdelyi), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

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B. R. Judd (1976) Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.

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External Links:: ISSN 0305-0041, Document, MathReview (Cataldo Agostinelli)
Referenced by:: §34.12.
Encodings:: BibTeX

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Jacobi’s Triple Product

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Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

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17.8.2

{{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}}.

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Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

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Bailey’s Bilateral Summations

►

17.8.4

{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

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K. W. J. Kadell (1994) A proof of the

q

-Macdonald-Morris conjecture for

BC_{n}

. Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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External Links:: ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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M. Katsurada (2003) Asymptotic expansions of certain

q

-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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External Links:: ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §17.2(i).
Encodings:: BibTeX

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D. E. Knuth (1992) Two notes on notation. Amer. Math. Monthly 99 (5), pp. 403–422.

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External Links:: ISSN 0002-9890, FullText, MathReview (W. Dörfler), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations, Notations.
Encodings:: BibTeX

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K. S. Kölbig (1986) Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17 (5), pp. 1232–1258.

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External Links:: ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §25.12(ii).
Encodings:: BibTeX

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T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

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R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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External Links:: ISSN 0022-4715, Document, MathReview, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

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B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:: ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §15.8(v), §20.11(iii).
Encodings:: BibTeX

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B. C. Berndt and R. J. Evans (1984) Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.

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External Links:: ISSN 0723-0869, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §2.10(iii).
Encodings:: BibTeX

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B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.

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External Links:: ISBN 0-387-96794-X, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §15.7.
Encodings:: BibTeX

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B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.

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External Links:: ISBN 0-387-97503-9, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §17.13.
Encodings:: BibTeX

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Ramanujan%E2%80%99s

21—30 of 615 matching pages

21: Bibliography H

22: Bibliography R

23: 5.13 Integrals

Ramanujan’s Beta Integral

24: 17.13 Integrals

Ramanujan’s Integrals

25: Bibliography L

26: Richard A. Askey

27: Bibliography J

28: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

Jacobi’s Triple Product

Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

Bailey’s Bilateral Summations

29: Bibliography K

30: Bibliography B

Ramanujan%E2%80%99s

21—30 of 615 matching pages

21: Bibliography H

22: Bibliography R

23: 5.13 Integrals

Ramanujan’s Beta Integral

24: 17.13 Integrals

Ramanujan’s Integrals

25: Bibliography L

26: Richard A. Askey

27: Bibliography J

28: 17.8 Special Cases of ψ r r Functions

Jacobi’s Triple Product

Ramanujan’s ψ 1 1 Summation

Bailey’s Bilateral Summations

29: Bibliography K

30: Bibliography B

28: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

Ramanujan’s ${{}_{1}\psi_{1}}$ Summation