Elliptic Functions

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 ?(?½).-
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.

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