# Hadamard, Jacques

### theory function proved mathematics

[adamah®] (1865–1963) French mathematician: developed theory of functionals.

Hadamard’s parents recognized his mathematical ability and he attended the École Normale Supérieure in Paris. His doctoral thesis was on function theory; he taught at the Lycée Buffon and then at Bordeaux. At 44 Hadamard became professor of mathematics at the Collège de France in Paris, and later at the École Polytechnique and École Centrale. In 1941, aged 76, he left occupied France for the USA and then joined the team in London using operational research for the RAF. Returning to France after the war, he retired to his interests in music, ferns and fungi.

Hadamard produced new insights in most areas of mathematics and influenced the development of the subject in many directions. He published over 300 papers containing novel and highly creative work. In the mid-1890s he studied analytic functions, that is those arising from a power series that converges. He proved the Cauchy test for convergence of a power series. In 1896 he proved the prime number theorem (first put forward by ) that the number of prime numbers less than *x* tends to *x* /log e *x* as *x* becomes large. This is the most important result so far discovered in number theory; it was independently proved by C J Poussin in the same year.

Hadamard investigated geodesics (or shortest paths) on surfaces of negative curvature (1888) and stimulated work in probability theory and ergodic theory. He then considered functions *f* ( *c* ) that depend on the path *c* , and defined a ‘functional’ *y* as *y* = *f* ( *c* ). The definitions of continuity, derivative and differential become generalizations of those for an ordinary function *y* = *f* ( *x* ) where *x* is just a variable. A new branch of mathematics, functional analysis, with relevance to physics and particularly quantum field theory grew out of this.

Hadamard also analysed functions of a complex variable and defined a singularity as a point at which the function is no longer regular. A set of singular points may still allow the function to be continuous–and such regions are called ‘lacunary space’, the subject of much modern mathematics. Finally he initiated the concept of a ‘well-posed problem’ as one in which a solution exists that is unique for the given data but depends continuously on those data. A typical example is the solution of a differential equation written as a convergent power series. This has proved to be a powerful and fruitful concept and since then the neighbourhood and continuity of function spaces have been studied. Hadamard published books on the psychology of the mathematical mind (on in particular) and was an inspiring lecturer who influenced several generations of mathematicians.

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