He spent a short while in the clerical service, another short while teaching in Annamalai University then, from 1931 to 1951, he taught in the Madras Christian College. Here he gained an outstanding reputation as a teacher of classical analysis.
In 1951 Rajagopal was persuaded to join the Ramanujan Institute of Mathematics then, four years later, he became head of the Institute. Under his leadership the Institute became the major Indian mathematics research centre.
Rajagopal studied sequences, series, summability. He published 89 papers in this area generalising and unifying Tauberian theorems.
He also studied functions of a complex variable giving an analogue of a theorem of Edmund Landau on partial sums of Fourier series. In several papers he studied the relation between the growth of the mean values of an entire function and that of its Dirichlet series.
A final topic to interest him was the history of medieval Indian mathematics. He showed that the series for tan-1x discovered by Gregory and those for sin x and cos x discovered by Newton were known to the Hindus 150 years earlier. He identified the Hindu mathematician Madhava as the first discoverer of these series.
Rajagopal is described in  as follows:-
Rajagopal was a teacher par excellence and a reliable and inspiring research guide. No words can adequately describe his modesty. Rational thinking and interest in psychic studies were two attributes which he imbibed with pride from his teacher Ananda Rau.
Article by: J J O'Connor and E F Robertson