04325nam 2200625 a 450 991077882440332120230802004439.01-280-37754-29786613555458981-4366-46-3(CKB)2550000000079949(EBL)840716(SSID)ssj0000598928(PQKBManifestationID)12199193(PQKBTitleCode)TC0000598928(PQKBWorkID)10591601(PQKB)11355927(MiAaPQ)EBC840716(WSP)00008252(Au-PeEL)EBL840716(CaPaEBR)ebr10524613(CaONFJC)MIL355545(OCoLC)877767895(EXLCZ)99255000000007994920120123d2012 uy 0engurbn|||||||||txtccrDevelopment of elliptic functions according to Ramanujan[electronic resource] /originally by K. Venkatachaliengar ; edited and revised by Shaun Cooper[Rev. ed.].Singapore ;Hackensack, N.J. World Scientificc20121 online resource (185 p.)Monographs in number theory,1793-8341 ;v. 6Originally published as a Technical Report 2 by Madurai Kamaraj University in February, 1988.981-4366-45-5 Includes bibliographical references and index.Preface; Contents; 1. The Basic Identity; 1.1 Introduction; 1.2 The generalized Ramanujan identity; 1.3 The Weierstrass elliptic function; 1.4 Notes; 2. The Differential Equations of P, Q and R; 2.1 Ramanujan's differential equations; 2.2 Ramanujan's 1ψ1 summation formula; 2.3 Ramanujan's transcendentals U2n and V2n; 2.4 The imaginary transformation and Dedekind's eta-function; 2.5 Notes; 3. The Jordan-Kronecker Function; 3.1 The Jordan-Kronecker function; 3.2 The fundamental multiplicative identity; 3.3 Partitions; 3.4 The hypergeometric function 2F1(1/2, 1/2; 1; x): first method3.5 Notes 4. The Weierstrassian Invariants; 4.1 Halphen's differential equations; 4.2 Jacobi's identities and sums of two and four squares; 4.3 Quadratic transformations; 4.4 The hypergeometric function 2F1(1/2, 1/2; 1; x): second method; 4.5 Notes; 5. The Weierstrassian Invariants, II; 5.1 Parameterizations of Eisenstein series; 5.2 Sums of eight squares and sums of eight triangular numbers; 5.3 Quadratic transformations; 5.4 The hypergeometric function 2F1(1/4, 3/4; 1; x); 5.5 The hypergeometric function 2F1(1/6, 5/6; 1; x); 5.6 The hypergeometric function 2F1(1/3, 2/3; 1; x)5.7 Notes 6. Development of Elliptic Functions; 6.1 Introduction; 6.2 Jacobian elliptic functions; 6.3 Reciprocals and quotients; 6.4 Derivatives; 6.5 Addition formulas; 6.6 Notes; 7. The Modular Function λ; 7.1 Introduction; 7.2 Modular equations; 7.3 Modular equation of degree 3; 7.4 Modular equation of degree 5; 7.5 Modular equation of degree 7; 7.6 Modular equation of degree 11; 7.7 Modular equation of degree 23; 7.8 Notes; Appendix A Singular Moduli; A.1 Notes; Appendix B The Quintuple Product Identity; B.1 Notes; Appendix C Addition Theorem of Elliptic Integrals; Bibliography; IndexThis unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter. The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan's work. It can be read by anyone with some undergraduate knowledge ofMonographs in number theory ;v. 6.Elliptic functionsElliptic functions.515.983Venkatachaliengar K1470934Cooper Shaun767406MiAaPQMiAaPQMiAaPQBOOK9910778824403321Development of elliptic functions according to Ramanujan3683018UNINA