About the Project

Ramanujan integrals

AdvancedHelp

(0.002 seconds)

1—10 of 22 matching pages

1: 5.13 Integrals
Ramanujan’s Beta Integral
2: 17.13 Integrals
Ramanujan’s Integrals
3: Bibliography P
  • P. I. Pastro (1985) Orthogonal polynomials and some q -beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
  • 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 π to high precision (Borwein and Borwein (1987, p. 26)). …
    5: Bibliography B
  • 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.
  • 6: 24.7 Integral Representations
    §24.7 Integral Representations
    §24.7(i) Bernoulli and Euler Numbers
    §24.7(ii) Bernoulli and Euler Polynomials
    Mellin–Barnes Integral
    For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2000, Chapters 3 and 4).
    7: 20.11 Generalizations and Analogs
    §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 integrals19.2(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
    §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 ) / ( π ( 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
    9: Bibliography L
  • D. H. Lehmer (1943) Ramanujan’s function τ ( n ) . Duke Math. J. 10 (3), pp. 483–492.
  • D. H. Lehmer (1947) The vanishing of Ramanujan’s function τ ( n ) . Duke Math. J. 14 (2), pp. 429–433.
  • 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.
  • Y. L. Luke (1968) Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.
  • Y. L. Luke (1970) Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.
  • 10: Bibliography D
  • B. Davies (1984) Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
  • L. Debnath and D. Bhatta (2015) Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.
  • A. Dienstfrey and J. Huang (2006) Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.
  • H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.
  • J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.