**Harry Bateman**'s parents were Marnie Elizabeth Bond and Samuel Bateman, who was a pharmaceutical chemist and commercial traveller. Harry attended Manchester Grammar School where he first grew to love mathematics, and in his final year he won a scholarship to Trinity College, Cambridge. He went up to Cambridge in 1900 and was Senior Wrangler in the Mathematical Tripos examinations of 1903 when he was awarded his B.A. (being Senior Wrangler means that he was ranked first among those students who were awarded a First Class degree). Bateman was awarded a Smith's prize in 1905 for an essay on differential equations. That year he became a Fellow of Trinity College and he received his M.A. degree in 1906. Some insight into the influence on his early mathematical development by his teachers at Cambridge is gained through a letter he wrote in July 1945 (quoted by Erdélyi in [3]):-

Bateman began publishing mathematics while he was still an undergraduate. He published eight problems in theMy own interest in the integrals of the Euler-Laplace type dates, I think, from the time when Sir Edmund Whittaker gave some properties of the Laplace transformation in his lectures at Cambridge in1903or1904. I made some use of the method of the inverse Laplace transformation in the Smith's prize essay and fellowship dissertation partly published in modified form in1909and1910.

*Educational Times*, between 1902 and 1910, and of these seven were on geometry. His first paper

*Determination of curves satisfying given conditions*was written while he was still an undergraduate and it was published in the

*Proceedings of the Cambridge Philosophical Society*in 1903. Two further papers appeared in print in 1904, namely

*The solution of partial differential equations by means of definite integrals*, and

*Certain definite integrals and expansions connected with the Legendre and Bessel functions*.

During the years 1905 and 1906 Bateman travelled on the continent, visiting Paris and Göttingen. It was during his visit to Göttingen that he learnt of work on integral equations being undertaken by Hilbert and his school. This prompted Bateman to use the methods being developed there to study the propagation of earthquake waves. On his return to England in 1906 he was appointed a lecturer at Liverpool University, becoming a Reader in mathematical physics at the University of Manchester in the following year. Over these years his publication record was amazing. In addition to the papers mentioned above, he published two papers in 1905, eight in 1906, five in 1907 and seven in 1908. One of these 1908 papers is his first publication on transformations of partial differential equations and their general solutions. This was to be a topic to which he made many important contributions, and we refer again to this work below. His 1908 paper was on the wave equation.

Bateman emigrated to the United States in 1910 having published four papers in 1909 and seventeen papers in 1910. The years 1910 -1912 were spent at Bryn Mawr College where Charlotte Angas Scott was the Head of Mathematics. He then spent the years 1912 - 1917 at Johns Hopkins University in Baltimore on a research fellowship. Frank Morley was the Head of Department at Johns Hopkins and by this stage he had restored the doctoral programme there which had been in a bad way when he took over in 1900. The appointment of Bateman was a brilliant move to bring an extremely active young research mathematician into the College. The year 1912 was the one during which Bateman married Ethel Horner Dodd on 11 July. They had one son Harry Graham who died while he was only a child, and they adopted a daughter Joan Margaret.

There is one strange aspect to Bateman's career which will not be evident from our description above. In 1913, while at Johns Hopkins University, he was awarded a Ph.D. At this time he was an extremely eminent mathematician with over 60 publications to his name, some of great importance. He did not have the typical CV of a Ph.D. student! His doctoral dissertation was entitled *The Quartic Curve and Its Inscribed Configurations* and his thesis supervisor was Frank Morley. However, while at Johns Hopkins he found time to take on jobs for a variety of different organisations, being employed at the Bureau of Standards, the Mount Saint Agnes College (a Baltimore college for women operated under the auspices of the Sisters of Mercy), the Weather Bureau, and the Department of Terrestrial Magnetism.

He spent the rest of his life at Pasadena after being appointed in 1917 as professor in the Division of Mathematics, Theoretical Physics, and Aeronautics at Throop College, which was renamed the California Institute of Technology about three years later. In fact when Bateman arrived there in 1917 the College had gone through a marked transition. Originally named Throop University when it was founded by Amos Throop in 1891, it became Throop Polytechnic Institute in 1893, then Throop College of Technology in 1913. The transformation had come about largely because of the enthusiasm of George Ellery Hale, the astronomer, to make the College into a leading scientific institution. A number of superb appointments, such as that of Bateman and also the physicist Robert A Millikan, helped achieve Hale's aim.

Some of Bateman's early work was on geometry and the influence of geometry on all his work is evident. In 1905 he studied certain quartic surfaces examined earlier by Cayley and Chasles. In particular he constructed the tangent plane and exhibited the surface as an envelope of planes. He is especially known for his work on special functions and partial differential equations. In 1904 he extended Whittaker's solution of the potential and wave equation by definite integrals to more general partial differential equations.

Bateman was one of the first to apply Laplace transforms to integral equations in 1906. In 1910 he solved systems of differential equations discovered by Rutherford which describe radio-active decay. Bateman's method was the now familiar one of applying the complex inversion formula of the Laplace transform. The finest contribution Bateman made to mathematics, however, was his work on transformations of partial differential equations, in particular his general solutions containing arbitrary functions. In particular he applied his methods to equations resulting from electromagnetics, then later to those arising from hydrodynamics.

He accumulated a vast store of information on all the familiar special functions and on his death the publication of his manuscripts was undertaken by Erdélyi and his associates in the form of the well-known series *Higher Transcendental Functions* and *Tables of Integral Transforms*. Copson writes:-

He wrote a number of texts that have been reprinted as classics:The late Harry Bateman, during his last years, planned an extensive compilation of the "Special Functions". He intended to investigate and tabulate their properties, the inter-relations between them, their representations in various forms, their macro- and micro-scopic behaviour, and to construct tables of the definite integrals involving them. The whole project was to have been on a gigantic scale; it would have been an authoritative and definitive account of its vast subject.While much of the material is available, it is not readily accessible, being scattered in books and journals on many fields. The "Guide to the Functions" which Bateman planned would have been invaluable. The project was never completed, and, after his death, the California Institute of Technology and the U. S. Office of Naval Research pooled their resources to continue Bateman's task.

It turned out that no single section of Bateman's work was in a state suitable for immediate publication, and the field was so wide that it appeared essential to narrow it down if anything useful was to be accomplished. It was decided to concentrate on a three-volume work on the Higher transcendental functions, to be followed by two volumes of tables of integrals. The whole work has been carried out by the staff of the Bateman Manuscript Project, under the directorship of Arthur Erdélyi.

*The mathematical analysis of electrical and optical wave-motion on the basis of Maxwell's equations*(1915, reprinted 1955);

*Partial differential equations of mathematical physics*(1932, reprinted 1944 and 1959); (written with H L Dryden and F D Murnaghan),

*Hydrodynamics*, National Research Council, Washington, D.C. (1932, reprinted 1956); and (written with A A Bennett and W E Milne),

*Numerical integration of differential equations*(1933, reprinted 1956). In all Bateman published around 200 papers in a period of 40 years. He only published five joint papers, one of those in 1924 being with Ehrenfest in which they looked at applications of partial differential equations to electromagnetic fields. Murnaghan wrote that his:-

Bateman received many honours for his contributions, including election to the Royal Society of London in 1928 and election to the National Academy of Sciences in Washington in 1930. He was elected vice-president of the American Mathematical Society in 1935, and was the Society's Gibbs lecturer in 1943. In fact he was on his way to New York to receive further recognition in the form of an award from the Institute of Aeronautical Science when he died of a coronary thrombosis. He was on a train near Milford, Utah, at the time of his death.... books and papers bristle with references which are a veritable mine of useful source material.

His interests outside mathematics were few (publishing 5 papers a year for 40 years does not leave much time over!). He was a top class chess player, however, good enough to represent Britain in a match against the USA when he was an undergraduate at Cambridge. He played in other chess tournaments too with some notable victories over leading players. His only other hobby appears to have been motoring.

Erdélyi writes in [3]:-

By all who knew him Bateman is praised as a charmingly modest and unassuming person, very quiet, almost of retiring disposition, but always helpful and ready to put his time and extensive knowledge at the disposal of others.

**Article by:** *J J O'Connor* and *E F Robertson*

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