Elliptic functions

K. Chandrasekharan

[目次]

  • I. Periods of meromorphic functions.- x 1. Meromorphic functions.- x 2. Periodic meromorphic functions.- x 3. Jacobi's lemma.- x 4. Elliptic functions.- x 5. The modular group and modular functions.- Notes on Chapter I.- II. General properties of elliptic functions.- x1. The period parallelogram.- x 2. Elementary properties of elliptic functions.- Notes on Chapter II.- III. Weierstrass's elliptic function ?(z).- x1. The convergence of a double series.- x 2. The elliptic function ?(z).- x 3. The differential equation associated with ?(z).- x 4. The addition-theorem.- x 5. The generation of elliptic functions.- Appendix I. The cubic equation.- Appendix II. The biquadratic equation.- Notes on Chapter III.- IV. The zeta-function and the sigma-function of Weierstrass.- x 1. The function ?(z).- x2. The function ?(z).- x 3. An expression for elliptic functions.- Notes on Chapter IV.- V. The theta-functions.- x1. The function ?(?, ?).- x 2. The four sigma-functions.- x 3. The four theta-functions.- x 4. The differential equation.- x 5. Jacobi's formula for ?' (0, ?).- x 6. The infinite products for the theta-functions.- x 7. Theta-functions as solutions of functional equations.- x 8. The transformation formula connecting ?3(v, ?) and ?3(?, ?1/?) ..- Notes on Chapter V.- VI. The modular function J(?).- x 1. Definition of J(?).- x 2. The functions g2(?) and g3(?).- x 3. Expansion of the function J(?) and the connexion with theta-functions.- x 4. The function J(?) in a fundamental domain of the modular group ..- x 5. Relations between the periods and the invariants of ?(u).- x 6. Elliptic integrals of the first kind.- Notes on Chapter VI.- VII. The Jacobian elliptic functions and the modular function ?(?).- x 1. The functions sn u, en u, dn u of Jacobi.- x 2. Definition by theta-functions.- x 3. Connexion with the sigma-functions.- x 4. The differential equation.- x 5. Infinite products for the Jacobian elliptic functions.- x 6. Addition-theorems for sn u, cn u, dn u.- x 7. The modular function ?(?).- x8. Mapping properties of ?(?) and Picard's theorem.- Notes on Chapter VII.- VIII. Dedekind's ?-function and Euler's theorem on pentagonal numbers.- x 1. Connexion with the invariants of the ?-function and with the theta-functions.- x 2. Euler's theorem and Jacobi's proof.- x 3. The transformation formula connecting ?(z) and ?(?1/2).- x4. Siegel's proof of Theorem 1.- x5. Connexion between ?(z) and the modular functions J(z), ?(z).- Notes on Chapter VIII.- IX. The law of quadratic reciprocity.- x 1. Reciprocity of generalized Gaussian sums.- x 2. Quadratic residues.- x3. The law of quadratic reciprocity.- Notes on Chapter IX.- X. The representation of a number as a sum of four squares ..- x1. The theorems of Lagrange and of Jacobi.- x 2. Proof of Jacobi's theorem by means of theta-functions.- x3. Siegel's proof of Jacobi's theorem.- Notes on Chapter X.- XI. The representation of a number by a quadratic form.- x1. Positive-definite quadratic forms.- x 2. Multiple theta-series and quadratic forms.- x 3. Theta-functions associated to positive-definite forms.- x 4. Representation of an even integer by a positive-definite form.- Notes on Chapter XI.- Chronological table.

「Nielsen BookData」より

[目次]

  • I. Periods of meromorphic functions.- x 1. Meromorphic functions.- x 2. Periodic meromorphic functions.- x 3. Jacobi's lemma.- x 4. Elliptic functions.- x 5. The modular group and modular functions.- Notes on Chapter I.- II. General properties of elliptic functions.- x1. The period parallelogram.- x 2. Elementary properties of elliptic functions.- Notes on Chapter II.- III. Weierstrass's elliptic function ?(z).- x1. The convergence of a double series.- x 2. The elliptic function ?(z).- x 3. The differential equation associated with ?(z).- x 4. The addition-theorem.- x 5. The generation of elliptic functions.- Appendix I. The cubic equation.- Appendix II. The biquadratic equation.- Notes on Chapter III.- IV. The zeta-function and the sigma-function of Weierstrass.- x 1. The function ?(z).- x2. The function ?(z).- x 3. An expression for elliptic functions.- Notes on Chapter IV.- V. The theta-functions.- x1. The function ?(?, ?).- x 2. The four sigma-functions.- x 3. The four theta-functions.- x 4. The differential equation.- x 5. Jacobi's formula for ?' (0, ?).- x 6. The infinite products for the theta-functions.- x 7. Theta-functions as solutions of functional equations.- x 8. The transformation formula connecting ?3(v, ?) and ?3(?, ?1/?) ..- Notes on Chapter V.- VI. The modular function J(?).- x 1. Definition of J(?).- x 2. The functions g2(?) and g3(?).- x 3. Expansion of the function J(?) and the connexion with theta-functions.- x 4. The function J(?) in a fundamental domain of the modular group ..- x 5. Relations between the periods and the invariants of ?(u).- x 6. Elliptic integrals of the first kind.- Notes on Chapter VI.- VII. The Jacobian elliptic functions and the modular function ?(?).- x 1. The functions sn u, en u, dn u of Jacobi.- x 2. Definition by theta-functions.- x 3. Connexion with the sigma-functions.- x 4. The differential equation.- x 5. Infinite products for the Jacobian elliptic functions.- x 6. Addition-theorems for sn u, cn u, dn u.- x 7. The modular function ?(?).- x8. Mapping properties of ?(?) and Picard's theorem.- Notes on Chapter VII.- VIII. Dedekind's ?-function and Euler's theorem on pentagonal numbers.- x 1. Connexion with the invariants of the ?-function and with the theta-functions.- x 2. Euler's theorem and Jacobi's proof.- x 3. The transformation formula connecting ?(z) and ?(?1/2).- x4. Siegel's proof of Theorem 1.- x5. Connexion between ?(z) and the modular functions J(z), ?(z).- Notes on Chapter VIII.- IX. The law of quadratic reciprocity.- x 1. Reciprocity of generalized Gaussian sums.- x 2. Quadratic residues.- x3. The law of quadratic reciprocity.- Notes on Chapter IX.- X. The representation of a number as a sum of four squares ..- x1. The theorems of Lagrange and of Jacobi.- x 2. Proof of Jacobi's theorem by means of theta-functions.- x3. Siegel's proof of Jacobi's theorem.- Notes on Chapter X.- XI. The representation of a number by a quadratic form.- x1. Positive-definite quadratic forms.- x 2. Multiple theta-series and quadratic forms.- x 3. Theta-functions associated to positive-definite forms.- x 4. Representation of an even integer by a positive-definite form.- Notes on Chapter XI.- Chronological table.

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この本の情報

書名 Elliptic functions
著作者等 Chandrasekharan, K.
Chandrasekharan Komaravolu
Chandrasekharan K
シリーズ名 Die Grundlehren der mathematischen Wissenschaften
出版元 Springer-Verlag
刊行年月 c1985
版表示 Softcover reprint of the original 1st ed. 1985
ページ数 xi, 189 p.
大きさ 24 cm
ISBN 0387152954
9783642522468
NCID BA0009917X
※クリックでCiNii Booksを表示
言語 英語
出版国 ドイツ
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