From Glasgow Carslaw went to Emmanuel College, Cambridge, where he graduated in 1894. Returning to the University of Glasgow as a lecturer in 1896, he made a visit to Göttingen during session 1896-97, where he worked with Sommerfeld, as well as visiting Rome and Palermo.
In 1903 Carslaw, then 33 years old, moved from his native Scotland to Australia where he had been offered the Chair of Mathematics at the University of Sydney. He had some impressive supporters. Thomson described his teaching as follows:-
His zeal and high acquirements as a mathematician, and his personal qualities, render him, in my opinion, remarkably well fitted for mathematical teaching in universities ...Thomson also said he was:-
... an enthusiast in original research, and having studied the mathematical papers and memoirs bearing on Fourier's series and their application in mathematical physics, purposes writing a book on the subject.It is doubtful whether his research record would put him in line for a chair today since before taking up the chair in Sydney he had published only four papers. However, he published two important books within three years of being appointed to Sydney. One was An introduction to the infinitesimal calculus published in 1905. Deakin remarks in  that the book was probably influenced by Hardy's lectures, saying:-
... it would be a brave historian indeed who saw Carslaw's 'little book' as being better than Hardy's tome, and a downright foolish one to claim it as more influential, nevertheless it did come first.The second book was Introduction to the theory of Fourier's series and integrals and the mathematical theory of the conduction of heat. This was to be the main area of Carslaw's research throughout his life. Jaeger in ,  and  claims this book to be Carslaw's most important contribution but Deakin in  claims it to be his later work on Laplace transforms. The fact that Jaeger himself collaborated with Carslaw on the Laplace transform work may explain why there are differing opinions here.
Carslaw married Ethel Maude Clarke from Rupertswood, Victoria, in 1907 but, sadly, she died within a year of their marriage.
Jaeger and Carslaw published Operational methods in applied mathematics in 1941. This put Heaviside's operational calculus on a rigorous footing following the approach proposed by Gustav Doetsch. Deakin writes in :-
In 1935, the Laplace transform was a topic of frontline research, by 1955 it was standard fare in undergraduate courses. No other advance has achieved such ready acceptance, and Carslaw and Jaeger's text can take a great deal of the credit.In fact this text was published six years after Carslaw retired. His final work published in 1947 was on income tax scales, one of the interests he had throughout his life :-
In his old age he interested himself in formulas designed to help in the just and efficient collection of income tax.Other topics to interest Carslaw throughout his career, which we have not touched on above, included an interest in non-euclidean geometry, Green's functions and the history of Napier's logarithms.
Article by: J J O'Connor and E F Robertson