# Cadambathur Tiruvenkatacharlu Rajagopal

### Quick Info

Triplicane, Madras, (now Chennai) India

Madras, (now Chennai) India

**Cadambathur Tiruvenkatacharlu Rajagopal**was an Indian mathematician who worked on three main areas: (i) sequences, series and summability; (ii) functions of a complex variable; and (iii) the history of medieval Kerala mathematics.

### Biography

**C T Rajagopal**'s parents were Cadambathur Tiruvenkatacharlu and Padmamma. Cadambathur Tiruvenkatacharlu worked in the Revenue Department of the Madras State. C T Rajagopal (we'll call him Rajagopal although he was known as CTR or Raju) had two younger brothers and one sister, the youngest of the four siblings. Rajagopal's eldest brother was C T Venugopal (1907-1972), known as Venu, who became an Indian civil servant who was recruited by the Indian Railway Accounts Service in 1930 and worked on the division of the assets of the railway when India and Pakistan became separate countries in 1947. Rajagopal's younger brother was Cadambur Tiruvenkatachari Krishnamachari Chari (1909-1993) who studied philosophy and became Head of the Department of Philosophy at Madras Christian College. He wrote on the theory of relativity, quantum mechanics and the paranormal. Rajagopal's youngest sibling was Kamala who married R P Sarathy, who became the first defence secretary after India after the split with Pakistan.

Rajagopal was educated in Madras, (now Chennai) India, where he attended the Hindu High School. This school began in 1860 when two boys schools in the Triplicane area of Madras were merged. The name Hindu High School was adopted in 1897 after a large new building was constructed. Rajagopal excelled in his studies at this school and, after his schooling was complete, he passed the Matriculation examination at Madras Presidency College in 1919. This College, founded in 1840, became part of the University of Madras when it was founded in 1857. Rajagopal first followed the Intermediate Course and passed the Intermediate Examination in 1921 with high marks. In the same year he began his study of the B.A. Honours Mathematics course but illness meant that he had to postpone taking the final examinations until 1925. He graduated in 1925 from the Madras Presidency College with First Class Honours in mathematics.

The Professor of Mathematics at Madras Presidency College when Rajagopal studied there was K Ananda Rau (1893-1966) who was an important influence on Rajagopal's mathematical development. Ananda Rau had studied at the University of Cambridge under G H Hardy who wrote on 26 March 1918 [3]:-

Mr K A Rau has spent the last year in mathematical research, undertaken in part under my advice. Some of his results are contained in an essay which obtained the Smith's Prize this year; and he has done other valuable work which will no doubt be published in due course.Ananda Rau was appointed Professor of Mathematics at Madras Presidency College in July 1919, the year in which Rajagopal began his studies at the College. M S Rangachari writes about Rajagopal's undergraduate studies in [5]:-

It was during the B.A. (Honours) course that he came into contact with K Ananda Rau, who was Professor of Mathematics in the Presidency College. The inspiration which Rajagopal got from Ananda Rau lasted throughout his life. In fact, there is an interesting parallel between the careers of these two men, which begins even earlier than their first contact, viz., their highly proficient studentship in the two renowned institutions, the Hindu High School and the Madras Presidency College. Ananda Rau could notice the exceptional intelligence of young Rajagopal. It is noteworthy that Tirukkannapuram Vijayaraghavan (1902-1955), who influenced Rajagopal in his later career, was a senior college-mate in the Honours classes of Rajagopal.In the year that he graduated, 1915, Rajagopal married Rukmini; their only son was C R Srinivasan.

Rajagopal spent over a year in clerical service at the Madras Accountant General's office, and another year 1930-31 teaching in Annamalai University. This university had only been founded in 1929. From 1931 to 1951, he taught in the Madras Christian College. Here he gained an outstanding reputation as a teacher of classical analysis [5]:-

It is here that his lucid and often original exposition of classical mathematical analysis and his tender care for students, in which he was in full measure supported by his wife, Rukmini ..., won for him their everlasting admiration, affection and respect. He built a house at Tambaram according to his own design and settled down, as he thought at that time, and named his abode "The Anchorage," teaching mathematics and doing research. Unlike many of Ananda Rau's students, Rajagopal started his research work late, some 14 years after his graduation. His initial papers were a prelude to what he was to take up for research and reflected his original exposition of topics which he taught for the honours classes.In fact his first paper,

*An integral test for the convergence of a series of positive terms*, was published in

*The Mathematics Student*. This journal of the Indian Mathematical Society had been founded in 1933 and was, for many years, publishing material which promoted the teaching of mathematics. Rajagopal's first research level publications appeared in 1937.

For a list of all of Rajagopal's publications, see THIS LINK.

In 1951 Rajagopal was persuaded to join the Ramanujan Institute of Mathematics by Tirukkannapuram Vijayaraghavan who was the first Director of the Institute which had been founded in 1950 by Sir Alagappa Chettiar, an Industrialist and Philanthropist. Four years later, after the death of Vijayaraghavan, Rajagopal became the Director of the Institute. This only happened after leading mathematicians André Weil, Ralph Boas and Antoni Zygmund had been consulted and advised on Rajagopal's appointment. Rajagopal's task as Director was, however, a very difficult one [5]:-

... the Institute passed through many a crisis and was often threatened with closure. That it survived those gloomy and agonising days was as much due to the high quality of mathematical activity generated by Rajagopal as it was to the indefatigable persistence, faith and hope with which he went on representing its cause to the authorities. ... when the fate of the Institute hung on the balance, the intervention by the famous astrophysicist Professor S Chandrasekhar made the then Prime Minister Jawaharlal Nehru make a recurring grant to the Institute and transfer its management to the University of Madras. The Institute retained its separate identity till 1967 when it became enlarged, with the University Department of Mathematics merging with it, to become the Ramanujan Institute for Advanced Study in Mathematics. Rajagopal's contribution to the Institute attaining the supreme position as one of the most important centres of mathematical research in India was immeasurable.In 1967 Rajagopal reached retirement age but his services were extended so that he could be continue as the Director of the extended Institute [5]:-

He finally retired from the Institute in June 1969 but continued his research work with a small financial assistance through the Institute. It is no exaggeration to say that the Institute, as it stands today, is a monument to the unremitting efforts of Rajagopal as it is to the memory of Srinivasa Ramanujan. It is also true that the Ramanujan Institute is finding it difficult to face, with courage, the assessment of its achievements because of the void created by Rajagopal's retirement. The research work of the Institute was during Rajagopal's time, so productive and significant that his passing away has created a vacuum in the country among active, hard analysts, of whom he was perhaps the last.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 [4]:-

Rajagopal took upon himself the task of establishing, to the satisfaction of mathematicians and mathematical historians, that the fundamentals of mathematical analysis were discovered and used in Kerala at least two centuries before Newton and Leibniz. He was assisted in this task by the late Prince Rama Varma of Cochin and his own students.He showed that the series for $\tan^{-1}x$ 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. D T Whiteside in his

*Mathematical Papers of Isaac Newton*(Cambridge, 1968) writes:-

These series for the sine and cosine, which here appear for the first time in a European manuscript, had, as we now know, already been displayed in a 1639 Malayalam compendium, the Yuktibhasa, which itself professes to be based on an early sixteenth-century Sanskrit original, the Tantrasangraha, perhaps the work of the little-known Hindu mathematician Nilakantha.Rajagopal is described in [1] 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.After he retired Rajagopal continued to undertake research both on summability and on medieval Kerala mathematics. In fact he published eighteen papers in the nine years of his retirement. He died of a "massive heart attack and without pain" in April 1978 [5]:-

Many who knew C T Rajagopal could not believe that he was no more on 25 April 1978. He was so healthy and was so regular in habits that nobody could believe that destiny would snatch him away so soon. The end was painless and his wife who was by his side could not believe that he had left her. A medical man, who was called by her, when her beloved husband stopped responding to her enquiries, declared him to have succumbed to a "massive heart-attack."M S Rangachari, the author of several of the biographies listed in the references below, was Rajagopal's student, then Rajagopal's colleague and collaborator. He writes a personal tribute in [5] of which we quote a part:-

Rajagopal was a teacher par excellence at any level, graduate, post-graduate or advanced. His teaching was marked by very detailed treatment of the topic, be it either real or complex analysis or statics or dynamics; it did not only adhere to the main text prescribed for the topic but covered additional ground throwing light on the teacher's original study of the topic. Thus he catered very well to the average students while inspiring the bright ones towards original work. ... As with his teacher, his way with research workers was to encourage and expect them to formulate their own problems and then to discuss the problems with them.

The author cannot forget the day when he was advised by Rajagopal to work on summability methods revising his own earlier suggestion to work in the theory of groups. It was the day when the author could generalise Rajagopal's results relating to Norlund summability. On the other hand, he was a very good collaborator who never distinguished his own contribution in a joint work from that of the others. Quite a few who were dedicated workers in other fields and who lacked support in their initial career were encouraged by Rajagopal. One such, Professor K Ramachandra, while unveiling the portrait of Rajagopal in the Ramanujan Institute could not suppress his emotion when he remembered the encouragement given to him by Rajagopal.

In the present day world of changing fashions in work in mathematics, Rajagopal never swerved from the principle of his revered teacher Ananda Rau, viz. to work in themes dear to us without caring for fads and fashions and to leave the assessment of the work to posterity.

...

Rajagopal wrote well; whether it was his mathematical papers, or his articles on Christianity or psychic studies or reviews or notices, he was very careful in the choice of idiom and of precise expression. Each one of the many letters he wrote to the author is a piece of literature.

...

Rajagopal was a believer in rational thinking. He did not take interest in religions rituals, but at the same time did not discourage others believing in them. During India's freedom struggle he demonstrated his nationalism by always wearing Khadi in response to the call of Mahatma Gandhi. About philosophy, unlike his friend Vijayaraghavan, he felt that every one should have his own philosophy. Vijayaraghavan's personal discussions with him about Visishtadvaita Philosophy and the Tamil songs of the Vaishnavite saints did not impress him. He more recently used to tell the author that he wished Vijayaraghavan were alive to discuss these matters with the author. On the other hand, Rajagopal believed in the scientific investigation of the so called supernormal and superhuman phenomena. He was quite proud of his views in this aspect, viz., what goes by the name of psychic research, and more particularly in the rebirth of the soul, and was happy that Srinivasa Ramanujan and Ananda Rau too had the same belief.

...

As one who had immense faith in the knowledge of the Indian seers and saints, Rajagopal remarked that it was but natural that these ideas were found in our own scriptures. Rajagopal had a valuable collection of literature pertaining to psychic research. His lighter reading material was high class detective stories.

Rajagopal was an exemplary gentleman. Nobody, knowing him closely has ever heard him raising his voice. If he ever became angry, it was always in defence of his colleagues and students. The Rajagopals were highly hospitable. The author and some of his colleagues cannot forget the long hours spent at their home. There was always a stream of visitors to their house in Tambaram where each one of the visitors was personally welcomed and treated by Rajagopal and his gracious wife Rukmini.

### References (show)

- C T Rajagopal,
*Bull. London Math. Soc.***13**(5) (1981), 451-458. - C T Rajagopal: September 8, 1903, to April 25, 1978,
*J. Anal.***1**(1993), vii. - C T Rajagopal, K Ananda R
*au, J. London Math.*S**oc**. 44 (1969), 1-6. - M S Rangachari, Prof. C T Rajag
*opal, Indian J.***Ma**th. 22 (1) (1980), i-xxix. - M S Rangachari, Cadambathur Tiruvenkatacharlu Rajagopal (1903
*-1978), Biographical Memoirs of fellows of the Indian National Science***A**cademy 8 (1984), 131-155. - Y Sitaraman, Professor C T Rajagopal (19
*03-1978), J. Math.*P**hy**s. Sci. 12 (5) (1978), i-xvi.

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update September 2020

Last Update September 2020