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E-grāmata: Elliptic Functions

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Elliptic FunctionsPalielināt
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This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

Recenzijas

"...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

Papildus informācija

Springer Book Archives
I. Periods of meromorphic functions.- §
1. Meromorphic functions.- §
2.
Periodic meromorphic functions.- §
3. Jacobis 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. Weierstrasss 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. Jacobis 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 Picards theorem.-
Notes on
Chapter VII.- VIII. Dedekinds ?-function and Eulers theorem on
pentagonal numbers.- §
1. Connexion with the invariants of the ?-function and
with the theta-functions.- §
2. Eulers theorem and Jacobis proof.- §
3. The
transformation formula connecting ?(z) and ?(?½).- §4. Siegels 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 Jacobis theorem by means of theta-functions.- §3.
Siegels proof of Jacobis 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.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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