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Inhaltsangabe
This book delves into the intricate world of elliptic functions, exploring their historical development, mathematical foundations, and diverse applications. The author guides readers through the contributions of renowned mathematicians like Abel, Jacobi, and Weierstrass, illuminating the evolution of elliptic function theory from its origins to its more advanced forms. The book skillfully contrasts different approaches, such as the classical methods of Legendre and Jacobi with the modern insights of Weierstrass and Riemann, providing a comprehensive understanding of the subject. It delves into the realm of complex analysis, examining concepts like Riemann surfaces and the nature of periodicity, showcasing the profound connections between elliptic functions and other branches of mathematics, including algebra, geometry, and number theory. Through a blend of theoretical rigor and practical applications, this book offers valuable insights into the world of elliptic functions. It serves as a testament to the enduring power and elegance of mathematical thought, revealing how these functions have shaped our understanding of diverse scientific fields and continue to inspire new discoveries.
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Excerpt from Lectures on the Theory of Elliptic Functions, Vol. 1
IN the publication of these lectures, it is proposed to present the Theory of Elliptic F unctions in three volumes, which are to include in general the following three phases of the subject.
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Reseña del editor
In the publication of these lectures, it is proposed to present the Theory of Elliptic Functions in three volumes, which are to include in general the following three phases of the subject: I. A nalysis; II. Applications toP roblems in Oeometry and Mechanics; III. General Arithmetic and Higher, A lgebra. In Volume I an attempt is made to give the essential principles of the theory. The elliptic functions considered as the inverse of the elliptic integrals have their origin in the immortal works of A bel and Jacobi. I have wished to treat from a philosophic, as well as from a formal standpoint, the existence, and as far as possible, the ultimate meaning of the functions introduced by these mathematicians, to discuss the theories which originated with them, to follow their development, and to extend as far as possible the principles which they established. In this development great assistance has been rendered by the works of Hermite, who contributed so much not only to the theory of elliptic functions but also to almost every form of mathematical thought. The theory of Weierstrass is studied side by side with the older theory, and the beautiful formulas which we owe to him are contrasted with the corresponding formulas of the earlier writers. Riemann introduced certain surfaces upon which he represented algebraic integrals, and by thus expressing his conceptions of anal3rtic functions he revealed a clearer insight into their meaning. Instead of generalizing either the theory of Jacobi or that of Weierstrass so as to embrace the whole subject, it is thought better to make these theories specializations of a more general theory. This general theory is treated by means of the Riemann surface, which at the same time shows the intimate relation between the two theories just mentioned. In Volume II a treatment of elliptic integrals is given. Here much attention is
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