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Maxime Bôcher's father was professor of modern languages at MIT so he came from a strong academic background. Bôcher studied at Harvard receiving his first degree from there in 1888. His course at Harvard was a broad one for, in addition to mathematics, he studied a remarkably broad range of topics including Latin, chemistry, philosophy, zoology, geography, meterology, art and music.
Bôcher then went to Göttingen where he attended lectures by Klein, Schönflies, Schwarz, Schur and Voigt. He was particularly attracted by Klein's course on Lamé functions which was given in session 188990. At Göttingen he also attended lecture courses by Klein on the potential function, on partial differential equation of mathematical physics and on noneuclidean geometry. He was awarded a doctorate for a dissertation on the Development of the Potential Function into Series and he was encouraged to study this by Klein.
Osgood, writing in [6] describes Bôcher's doctoral thesis in these terms:
Though the leading ideas had been set forth by Klein in his lectures, nothing could be further from the truth than to think that Bôcher merely elaborated some details. he subject was an exceedingly broad one. It required for its treatment not so much a specific knowledge of the theory of the potential, although Bôcher was thoroughly equipped on that side, even familiarity with the geometry of inversion, of which he made himself a master, but rather the power to carry through a piece of detailed analytic investigation with accuracy and skill...Bôcher returned to Harvard as an instructor, then in 1894 assistant professor. He became professor of mathematics in 1904.
Bôcher published around 100 papers on differential equations, series, and algebra. His work on algebra, published in 1907, was particularly important. In an article on Fourier series he gave the first satisfactory treatment of the Gibbs phenomenon. In [3] his papers are said to:
... excell in simplicity and elegance and nearly all of them treat subjects of great importance to marked advantage. He never occupied himself with an unimportant problem.Bôcher's books are singled out in [3] for special mention:
Bôcher's Introduction to Higher Algebra, translated into German and Russian, was a remarkable pioneer work in English, which was long of great service to students. ... Yet another exceptional service was rendered by his Introduction to the Study of Integral Equations ..., the emphasis placed on the historical development of the subject being an interesting feature of the tract. Special attention should be drawn also to his little known pamphlet on regular point of linear differential equations of the second order used for a number of years in connection with one of his courses of lectures. Because of the clarity and care with which his elementary texts on analytic geometry and trigonometry were written they are still in demand.Bôcher was honoured by the American Mathematical Society when he was chosen to give the first series of Colloquium lectures. He gave six lectures on Linear differential equations and their applications in 1906. He was the first editorinchief of the Transactions of the American Mathematical Society holding this post for five years. In 1912 Bôcher was an invited speaker at the International Congress of Mathematicians held in Cambridge, England. He lectured there on Boundary problems in one dimension.
References (6 books/articles)
References elsewhere in this archive:
Tell me about Bôcher's work on matrices and determinants
Maxime Bôcher was the President of the American Mathematical Society in 1909  1910. You can see a history of the AMS and a list of AMS presidents.
He was the American Mathematical Society Colloquium Lecturer in 1896. You can see a history of the AMS Colloquium and a list of the lecturers.