Johann Peter Gustav Lejeune Dirichlet (13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.
Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815.
Dirichlet decided to go to Paris in May 1822. There he attended classes at the Collège de France and at the Faculté des sciences de Paris, learning mathematics from Hachette among others, while undertaking private study of Gauss’s Disquisitiones Arithmeticae, a book he kept close for his entire life. In 1823 he was recommended to General Foy, who hired him as a private tutor to teach his children German, the wage finally allowing Dirichlet to become independent from his parents’ financial support.
- Number theory
Number theory was Dirichlet’s main research interest, a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837, he published Dirichlet’s theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions.
Dirichlet found and proved the convergence conditions for Fourier series decomposition. Pictured: the first four Fourier series approximations for a square wave.
Inspired by the work of his mentor in Paris, Dirichlet published in 1829 a famous memoir giving the conditions, showing for which functions the convergence of the Fourier series holds.
- Definition of function
While trying to gauge the range of functions for which convergence of the Fourier series can be shown, Dirichlet defines a function by the property that “to any x there corresponds a single finite y”, but then restricts his attention to piecewise continuous functions.
In the summer of 1858, during a trip to Montreux, Dirichlet suffered a heart attack. On 5 May 1859, he died in Göttingen, several months after the death of his wife Rebecka.Dirichlet’s brain is preserved in the department of physiology at the University of Göttingen, along with the brain of Gauss.[dubious – discuss] The Academy in Berlin honored him with a formal memorial speech presented by Kummer in 1860, and later ordered the publication of his collected works edited by Kronecker and Lazarus Fuchs.