Bôcher was awarded a Harvard Fellowship, a Harris Fellowship and a Parker Fellowship which allowed him to travel to Europe to undertake research. The leading university for mathematics was Göttingen and there 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 1889-90. At Göttingen he also attended lecture courses by Klein on the potential function, on partial differential equations of mathematical physics and on non-euclidean geometry. He was awarded a doctorate in 1891 for his dissertation Über die Reihenentwicklungen der Potentialtheorie having been encouraged to study this topic by Klein who acted as supervisor. It was an outstanding piece of work which received a university prize from Göttingen.
Osgood, writing in , 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. The 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 ...While Bôcher was in Göttingen he met Marie Niemann and they were married in July 1891 after Bôcher had submitted his doctoral thesis. The Bôchers had three children, Helen, Esther, and Frederick. He returned with his new wife to Harvard where he was appointed as an instructor. In 1892 he published five papers: On Bessel's functions of the second kind; On a nine-point conic; On some applications of Bessel's functions with pure imaginary index; Note on the nine-point conic; and Some propositions concerning the geometric representation of imaginaries. Given his impressive record it is not surprising that in 1894 he was promoted to assistant professor. In that year he published his first book which was an extended version of his thesis with the same title. Perhaps 'extended version' does not do it justice since this book was now four times the length of his doctoral thesis. He became a full professor of mathematics in 1904.
Bôcher published around 100 papers on differential equations, series, and algebra. His text on algebra Introduction to higher algebra, published by Macmillan, New York in 1907, was particularly important. In a seventy page article in 1906, Introduction to the theory of Fourier's series published in the Annals of Mathematics, he gave the first satisfactory treatment of the Gibbs phenomenon (he wrote another paper On Gibbs' phenomenon in 1914). In  his papers are said to:-
... excel 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.Let us mention, in particular, The theory of linear dependence which he published in the Annals of Mathematics in 1900. This paper treats both the algebraic and functional notions in a unified fashion.
Bôcher's books are singled out in  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.When An introduction to the study of integral equations was reprinted in 1971 a reviewer wrote:-
The original was the first connected account of the subject. It was written when the pioneering work of Volterra and Fredholm was still fresh in the memory and while Hilbert, Erhard Schmidt and Weyl were in full flood. It is remarkable that this tract, which can still be read as a text-book, conveys so well the excitement of the time.He also wrote elementary texts such as Trigonometry (written jointly with Gaylord) and Analytic geometry. His final book was Leçons sur les méthodes de Sturm dans la théorie des équations différentielles linéaires et leurs développements modernes (1917) which was a record of lectures he gave in Paris in 1913-14 when he was Harvard Exchange Professor at the University of Paris. Although he was only 46 years old when he spent the year in Paris there was already signs that his health, which had never been particularly strong, was failing. He died at his home in Cambridge, Massachusetts, after suffering a prolonged illness.
As to his character Osgood writes  (see also  which reproduces ):-
Above all, Bôcher was sincere. He liked to argue and to defend a position; but when the game was over, it was the truth which had been brought out that pleased him most. He distrusted popular conclusions, even when the public was a learned one. It was facts, not views, that he sought, and his own intellect was the final arbiter ...Later in the article Osgood writes:-
His nature was reserved. He would not talk on personal matters relating to himself and this disinclination extended even to his scientific work. He was, however, glad to discuss the work of others with them. He was quick to grasp the central idea and often could express it more clearly than its author.Bôcher was honoured by the American Mathematical Society when he was chosen to give the first series of Colloquium lectures in 1896. He gave six lectures on Linear differential equations and their applications. He was a founder and the first editor-in-chief of the Transactions of the American Mathematical Society holding this post for five years in total during two spell, the first beginning in 1908 and the second in 1911. He was honoured with election to the National Academy of Sciences (United States) in 1909 and he served as president of the American Mathematical Society during 1909-1910 delivering his presidential address in Chicago on The published and unpublished works of Charles Sturm on algebraic and differential equations. 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.
Zund, in , gives this assessment:-
... it is difficult to adequately appreciate Bôcher's influence on his time because so much of his work was devoted to perfecting and polishing material rather than to producing striking new results that would bear his name. Nevertheless, his instinct and sense of what was important was impressive and much of his work became commonplace knowledge although his authorship was largely forgotten. ... In style and temperament Bôcher was a consummate artist ...
Article by: J J O'Connor and E F Robertson