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All Titchmarsh's work is in analysis, in fact he refused to lecture on any other topic. His method of working was to work on a topic until he tired of it, when he would write a book on that topic. He studied Fourier series, Fourier integrals writing Introduction to the Theory of Fourier Integrals (1937) and, working with Hardy, integral equations.
He also did important work on the Riemann zeta function wroting The Zeta-Function of Riemann (1930) and The Theory of the Riemann Zeta-Function (1951). His most popular book The theory of functions was published in 1932. He was elected to the Royal Society in 1931.
From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics. It was published in Eigenfunction Expansions Associated with Second-Order Differential Equations (1946,1958).
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