Ramanujan integrals

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1: 17.13 Integrals

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Ramanujan’s Integrals

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2: 5.13 Integrals

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Ramanujan’s Beta Integral

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3: Bibliography P

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Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

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ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(iv), §18.33(v).

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4: 19.35 Other Applications

… ►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …

5: Bibliography B

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Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.

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ISSN 0723-0869, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.10(iii).

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6: 24.7 Integral Representations

§24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

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§24.7(ii) Bernoulli and Euler Polynomials

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Mellin–Barnes Integral

… ►For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2015, Chapters 3 and 4).

7: 20.11 Generalizations and Analogs

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§20.11(ii) Ramanujan’s Theta Function and $q$-Series

►Ramanujan’s theta function $f(a,b)$ is defined by … ►

§20.11(iii) Ramanujan’s Change of Base

►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). … ►These results are called Ramanujan’s changes of base. …

8: 19.9 Inequalities

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§19.9(i) Complete Integrals

… ►The earliest is due to Kepler and the most accurate to Ramanujan. Ramanujan’s approximation and its leading error term yield the following approximation to $L(a,b)/(\pi (a+b))$: …Barnard et al. (2000) shows that nine of the thirteen approximations, including Ramanujan’s, are from below and four are from above. … ►

§19.9(ii) Incomplete Integrals

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9: Bibliography L

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Ramanujan’s function $\tau (n)$ . Duke Math. J. 10 (3), pp. 483–492.

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Document, MathReview (H. S. Zuckerman), ZentralBlatt

Referenced by:

§27.20, §27.20, §27.21.

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The vanishing of Ramanujan’s function $\tau (n)$ . Duke Math. J. 14 (2), pp. 429–433.

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Document, MathReview (R. A. Rankin), ZentralBlatt

Referenced by:

§27.14(vi).

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

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.16.

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Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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10: Bibliography D

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Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.

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ISBN 0-387-96080-5, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

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Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.

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ISBN 978-1-4822-2357-6, MathReview Entry, ZentralBlatt

Referenced by:

§1.16(viii).

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Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.

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ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§23.11.

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

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Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.

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External Links:

MathReview (Alan Weinstein), ZentralBlatt

Referenced by:

§36.12(iii).

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