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Thomas Bromwich spent his youth in Natal and then received his schooling in Durban. He returned to England to study at St John's College Cambridge arriving in 1892, in the same year as Whittaker. He obtained a Fellowship in 1897 but left Cambridge to become professor of mathematics at Galway in 1902. He left Galway 1907 and returned to a permanent post as College lecturer at St John's College Cambridge.
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. Some of this work is described in  where its history is explained:-
... T J I'A Bromwich's method for solving the source-free Maxwell equations for electromagnetic waves. ... was originally used by Bromwich in 1899, and subsequently independently discovered by H M Macdonald. It was first included as a question in Part II of the Mathematical Tripos in 1910 and featured by Bromwich in his lectures at Cambridge during the same year. He published this method in 1919, and it also formed the basis of his seminal paper on the scattering of plane electric waves by spheres.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 which was based on lectures on analysis he had given at Galway. 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 1906 he published Quadratic Forms and their Classification by Means of Invariant Factors. Gow  writes:-
This book is an early example in English of the more abstract methods introduced into algebra by researchers such as Kronecker and Weierstrass. It is particularly concerned with the simultaneous reduction of two quadratic forms, a problem which, in its modern presentation, requires almost the full repertoire of the theory of a single linear transformation.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  :
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 he became afflicted by a mental disorder which eventually led to his suicide.
References (5 books/articles)
References elsewhere in this archive:
Thomas J L'A Bromwich was elected to the Royal Society of London in 1906. You can see a history of the Royal Society and a list of the members among the mathematicians in our archive.
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