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By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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Extra resources for A selection of problems in the theory of numbers

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Inspired by his colleagues, his first publications are in applied mathematics, and he remained interested in the subject all his life. Ironically, success came easily to him with almost every thing he touched except the topic that made his name internationally and which interests us most: elliptic functions. His friend in later life, Dirichlet, tells this story in his memorial address of Jacobi27: One of his friends who noticed him in a bad mood one day, received this answer when he asked why he was out of sorts: You see me on the point of returning this book (Legendre’s Exercises) to the library, with which I’ve been exceedingly unlucky.

6 For a recent discussion, see Ferraro (2008). 7 For various reasons and in various ways power series expansions were a legitimate and common device in the study of functions, albeit of a formal kind to modern eyes. They retained much of this character, for example, in the work of Jacobi. Power series methods were commonly used in the study of differential equations when more informative solution methods failed. There was, by 1800, an impressive literature on ordinary differential equations. Linear equations had been given an (admittedly formal) theory by Lagrange, and some, like the hypergeometric equation, had been the object of more detailed study (as will be seen below).

Legendre did more than write privately to Jacobi, he communicated his results to the Acad´emie in the warmest terms on 5 November 1827, and his report was published in the Globe on the 29th. Jacobi was delighted with the much older man’s response and took the opportunity of his reply on 12 January 1828 to acquaint Legendre with the content of Abel’s Recherches, the first part of which was now in print. He put Abel’s work into his own notation, but otherwise summarised it much as we have above, dwelling on the striking division of the lemniscate.

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A selection of problems in the theory of numbers by Waclaw Sierpinski, I. N. Sneddon, M. Stark

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