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Ramanujan's Notebooks

by Bruce C. Berndt

  • ISBN: 9780387949413
  • ISBN10: 0387949410

Ramanujan's Notebooks

by Bruce C. Berndt

  • List Price: $239.00
  • Binding: Hardcover
  • Publisher: Springer Verlag
  • Publish date: 12/01/1997
  • ISBN: 9780387949413
  • ISBN10: 0387949410
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Description: 32 Continued Fractions.- 1 The Rogers--Ramanujan Continued Fraction.- 2 Other q--Continued Fractions.- 3 Continued Fractions Arising from Products of Gamma Functions.- 4 Other Continued Fractions.- 5 General Theorems.- 33 Ramanujan's Theories of Elliptic Functions to Alternative Bases.- 1 Introduction.- 2 Ramanujan's Cubic Transformation, the Borweins' Cubic Theta--Function Identity, and the Inversion Formula.- 3 The Principles of Triplication and Trimidiation.- 4 The Eisenstein Series L, M, and N.- 5 A Hypergeometric Transformation and Associated Transfer Principle.- 6 More Higher Order Transformations for Hypergeometric Series.- 7 Modular Equations in the Theory of Signature 3.- 8 The Inversion of an Analogue of K (k) in Signature 3.- 9 The Theory for Signature 4.- 10 Modular Equations in the Theory of Signature 4.- 11 The Theory for Signature 6.- 12 An Identity from the First Notebook and Further Hypergeometric Transformations.- 13 Some Enigmatic Formulas Near the End of the Third Notebook.- 14 Concluding Remarks.- 34 Class Invariants and Singular Moduli.- 1 Introduction.- 2 Table of Class Invariants.- 3 Computation of Gnand gnwhen 9/n.- 4 Kronecker's Limit Formula and General Formulas for Class Invariants.- 5 Class Invariants Via Kronecker's Limit Formula.- 6 Class Invariants Via Modular Equations.- 7 Class Invariants Via Class Field Theory.- 8 Miscellaneous Results.- 9 Singular Moduli.- 10 A Certain Rational Function of Singular Moduli.- 11 The Modular j-invariant.- 35 Values of Theta-Functions.- 0 Introduction.- 1 Elementary Values.- 2 Nonelementary Values of.- 3 A Remarkable Product of Theta-Functions.- 36 Modular Equations and Theta-Function Identities in Notebook 1.- 1 Modular Equations of Degree 3 and Related Theta-Function Identities.- 2 Modular Equations of Degree 5 and Related Theta-Function Identities.- 3 Other Modular Equations and Related Theta-Function Identities.- 4 Identities Involving Lambert Series.- 5 Identities Involving Eisenstein Series.- 6 Modular Equations in the Form of Schlfli.- 7 Modular Equations in the Form of Russell.- 8 Modular Equations in the Form of Weber.- 9 Series Transformations Associated with Theta-Functions.- 10 Miscellaneous Results.- 37 Infinite Series.- 38 Approximations and Asymptotic Expansions.- 39 Miscellaneous Results in the First Notebook.- Location of Entries in the Unorganized Portions of Ramanujan's First Notebook.- References.
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