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During his tenure of the chair at Galway he was elected a Fellow of the Royal Society (1906). He was an active member of the London Mathematical Society, being Secretary from 1911 until 1919 when he was elected Vice-President.
Bromwich's first research was on applied mathematics where was influenced by Stokes. He made significant contributions to electromagnetism although he was always less interested in the physics, more in the mathematics.
Bromwich worked on infinite series, particularly during his time in Galway. In 1908 he published his only large treatise An introduction to the theory of infinite series. Hardy said
The book is unquestionably a very fine one. It is not merely a good and an interesting book: it has a character and a distinction which show at once that it is written by an exceptional mathematician.Bromwich also made useful contributions to quadratic and bilinear forms and many consider his algebraic work to be his finest.
In a series of papers he put Heaviside's calculus on a rigorous basis treating the operators as contour integrals.
Bromwich did his best work before reaching the age of 33 perhaps because of overwork after this time. Hardy said, see [2] :
He was engaged in original work in several different fields: he put a great deal of energy into his college and university lectures, where his passion for working out every detail must have added enormously to his labours: and to all this he added a considerable amount of examining and private coaching. ... He would have had a happier life, and been a greater mathematician , if his mind had worked with less precision.Unfortunately his health began to suffer and became afflicted by a mental disorder which eventually led to his suicide.
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