Ramanujan integrals

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11: Bibliography

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Chapter $16$ of Ramanujan’s second notebook: Theta-functions and $q$-series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.

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External Links:

ISSN 0065-9266, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§20.11(ii).

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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.9(i).

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

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Ramanujan Revisited. Academic Press Inc., Boston, MA.

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External Links:

ISBN 0-12-058560-X, ISBN 0-12-058560-X, MathReview, ZentralBlatt

Referenced by:

§20.11(ii), Profile Richard A. Askey.

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

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12: Bibliography K

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On an integral of Hardy and Littlewood. Kyushu J. Math. 52 (1), pp. 249–263.

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ISSN 1340-6116, Document, MathReview (Fritz Haslinger), ZentralBlatt

Referenced by:

§9.13(ii).

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Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§19.12.

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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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ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

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An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.

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ISSN 0044-2275, Document, MathReview, ZentralBlatt

Referenced by:

§10.71(ii).

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

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ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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13: 8.11 Asymptotic Approximations and Expansions

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8.11.11 $${\gamma }^{\ast }(1-a,-x)=x^{a-1}\left(-\cos \left(\pi a\right)+\frac{\sin \left(\pi a\right)}{\pi }\left(2\sqrt{\pi }F\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi }{a}}\left(1-y^2\right)\right){\mathrm{e}}^{y^2}+O\left(a^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E11

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Annotations for §8.11(iv), §8.11 and Ch.8

… ►For Dawson’s integral $F\left(y\right)$ see §7.2(ii). …

14: 17.2 Calculus

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§17.2(v) Integrals

►If $f(x)$ is continuous at $x=0$, then ►

17.2.45 $${\int }_0^{1}f(x)d_{q}x=(1-q)\sum _{j=0}^{\mathrm{\infty }}f(q^j)q^j,$$

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Symbols:

$\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.2.E45

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Annotations for §17.2(v), §17.2 and Ch.17

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17.2.47 $$\underset{q\to 1-}{lim}{\int }_0^{a}f(x)d_{q}x={\int }_0^{a}f(x)dx.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.2.E47

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Annotations for §17.2(v), §17.2 and Ch.17

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§17.2(vi) Rogers–Ramanujan Identities

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15: Bibliography C

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On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.

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External Links:

MathReview (R. Nicolovius), ZentralBlatt

Referenced by:

§19.36(ii), §19.36(ii).

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Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.

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External Links:

FullText, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.24(ii), §19.9(ii).

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On Ramanujan’s cubic transformation formula for ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};1;z)$ . Math. Proc. Cambridge Philos. Soc. 124 (2), pp. 193–204.

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ISSN 0305-0041, Document, MathReview (Aleksander Grytczuk), ZentralBlatt

Referenced by:

§15.8(v).

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Approximations and Complex Multiplication According to Ramanujan. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), G. E. Andrews, R. A. Askey, B. C. Bernd, K. G. Ramanathan, and R. A. Rankin (Eds.), pp. 375–472.

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External Links:

MathReview (Maurice Mignotte), ZentralBlatt

Referenced by:

§20.11(iii).

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Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§19.13(i).

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16: Bibliography S

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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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External Links:

Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

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Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:

Double-precision Fortran.

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FullText, Code

Referenced by:

5th item.

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On an identity of Ramanujan based on the hypergeometric series ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};\frac{1}{2};x)$ . J. Number Theory 69 (2), pp. 125–134.

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External Links:

ISSN 0022-314X, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§20.11(iii).

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Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:

Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

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The Use of Integral Transforms. McGraw-Hill, New York.

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External Links:

ISBN 00-705-9436-8, ZentralBlatt

Referenced by:

§1.14(v), §10.43(v), §14.31(ii).

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17: 27.21 Tables

… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\ln x$, and $\mathrm{li}\left(x\right)$. … ►Tables of the Ramanujan function $\tau \left(n\right)$ are published in Lehmer (1943) and Watson (1949). …

18: Bibliography R

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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).

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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).

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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).

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

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19: Bibliography H

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Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§17.13.

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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).

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Elliptic Integrals. Dover Publications Inc., New York.

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Notes:

Unaltered reprint of original edition published by Wiley, New York, 1917.

External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§19.8(ii).

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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).

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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).

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20: Bibliography W

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Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

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A table of Ramanujan’s function $\tau (n)$ . Proc. London Math. Soc. (2) 51, pp. 1–13.

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External Links:

Document, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.21.

Encodings:

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Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.22(vi), §10.23(iv), §10.43(vi).

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A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

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