Elliptic Functions by Komaravolu Chandrasekharan (PDF)

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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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a nuts and bolts book that deals with the complex analytic properties of elliptic functions and theta functions together with some of their applications to number theory. The material is all classical. Chandrasekharan is a careful writer, and he works out all the important details in the proofs that I have read. This includes verifying in detail the convergence of the various series and products with which this book deals. Using the machinery, he gives proofs of Picard’s little theorem, which is fairly standard and uses the modular function lambda, a proof of the law of quadratic reciprocity, and a proof of Jacobi’s formula for the number of representation of a number as a sum of four squares.Chandrasekharan concludes each of the chapters with detailed, perceptive, and factually correct historical notes.This book is also good for proofs of basic facts about modular forms, and for following the first stage in the development of the theory of modular forms. For more historical details and for an historically minded development of the theory of elliptic functions, one should read Weil’s 1976 “Elliptic functions according to Eisenstein and Kronecker”, and for more modern or advanced developments in modular forms, one should read the relevant chapters in Iwaniec and Kowalski’s “Analytic number theory”, which however is less carefully written than Chandrasekharan’s works.

⭐Let me start by saying that this book really is not worth the hefty price tag. I’m not sure why it’s so expensive.Elliptic Functions is a pretty cool subject. I got into this out of interest in elliptic curves and modular forms. However, the connection to those subjects is not very apparent in this book. Chandrasekharan’s book seems to focus on the early development of this theory (I’d guess from the late-1800’s to early 20th century). Focus is primarily on analytic foundations and connections to Analytic Number Theory.No exercizes are included.For more on Elliptic Functions in the context of modern mathematics, check out the book by Serge Lang.

⭐私が超弦理論の基礎を学習しているときに、θ関数を勉強する必要が生じましたが、非常にわかりやすい説明でθ関数の基礎を学習する上で非常に優れた教科書であると思います。特に、p-関数から始まってJacobi’s abstruse identityの導出を行うプロセスはこの本で勉強するのが一番いいでしょう。

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