Abelian Functions: Abel's Theorem and the Allied Theory of Theta Functions

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Classical algebraic geometry, inseparably connected with the names of Abel, Riemann, Weierstrass, Poincaré, Clebsch, Jacobi and other outstanding mathematicians of the last century, was mainly an analytical theory. In our century the methods and ideas of topology, commutative algebra and Grothendieck's schemes enriched it and seemed to have replaced once and forever the somewhat naive language of classical algebraic geometry. This classic book, written in 1897, covers the whole of algebraic geometry and associated theories. Baker discusses the subject in terms of transcendental functions, and theta functions in particular. Many of the ideas put forward are of continuing relevance today, and some of the most exciting ideas from theoretical physics draw on work presented here.

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Editorial Reviews

From the Publisher
"...offers welcome historical insight into a key point in the history of algebraic geometry." John B. Little, Mathematical Reviews
A reissue of the 1897 classic that despite its modest title, in fact covers the whole of algebraic geometry and associated theories. Some of the topics are the infinities of rational functions, specification of a general form of Riemann's integral, Jacobi's inversion problem, radical and factorial functions, the transformation of theta functions, and degenerate Abelian integrals. An added forward provides a historical context and suggests the continuing importance of the methods. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780521498777
  • Publisher: Cambridge University Press
  • Publication date: 12/28/2005
  • Series: Cambridge Mathematical Library Series
  • Edition description: REISSUE
  • Pages: 724
  • Product dimensions: 5.98 (w) x 8.98 (h) x 1.46 (d)

Table of Contents

1. The subject of investigation; 2. The fundamental functions on a Riemann surface; 3. The infinities of rational functions; 4. Specification of a general form of Riemann's integrals; 5. Certain forms of the fundamental equation of the Riemann surface; 6. Geometrical investigations; 7. Coordination of simple elements; 8. Abel's theorem; 9. Jacobi's inversion problem; 10. Riemann's theta functions; 11. The hyperelliptic case of Riemann's theta functions; 12. A particular form of Riemann surface; 13. Radical functions; 14. Factorial functions; 15. Relations concerning products of theta functions; 16. A direct method of obtaining the equations relating theta functions; 17. Theta relations with certain groups of characteristics; 18. Transformation of periods; 19. On systems of periods and on general Jacobian functions; 20. Transformation of theta functions; 21. Complex multiplication of theta functions; 22. Degenerate Abelian integrals; Appendix 1. On algebraic curves in space; Appendix 2. On matrices.

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