Elliptic Functions - Komaravolu Chandrasekharan

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

...In the breadth, depth and inevitability of treatment of this beautiful material, the author has made a contribution to the mathematical community consistent with the distinction of his career. That he has succeeded in compressing this treatment into a succinct monograph of fewer than 190 pages is a testament to his taste, discipline and powers of exposition.-- MATHEMATICAL REVIEWS