# Johann Peter Gustave Lejeune Dirichlet

# Johann Peter Gustave Lejeune Dirichlet

**1805-1859**

**German Mathematician**

Lejeune Dirichlet was a professor at the University of Berlin prior to accepting a chair—previously held by Carl Gauss (1777-1855)—at the University of Göttingen. Perhaps Dirichlet's most valuable contribution to science was in the theory of the Fourier series, which he originated. He also made important contributions to algebraic number theory and harmonic functions.

Dirichlet was born in Düren, then part of the French Empire and now part of Germany. The first mention of his work appears in 1826, when he proved one of Gauss's conjectures about prime numbers. The next year he was awarded a teaching position at the University of Breslau, after which he moved to the University of Berlin in 1828 and, in 1855, to Göttingen.

It was during his time at Berlin that Dirichlet first proposed what is the currently accepted definition of a function: If a variable *y* is so related to a variable *x* that whenever a numerical value is assigned to *x,* there is a rule according to which a unique value of *y* is determined, then *y *is said to be a function of the independent variable *x.*

In this definition, the key concept is that, for each number *x,* there is a unique value for *y, *something that had not previously been stated so simply. Or, put another way, any line drawn that is parallel to the *y*-axis will pass through the function no more than one time.

Dirichlet is best known, however, for his papers on the convergence of trigonometric series toward a solution, and in the use of such series to represent arbitrary functions. This work applied especially well to the Fourier series. The previous work of Simeon Poisson (1781-1840) on the Fourier series was shown by Augustin Cauchy (1789-1857) to be inadequate, but Dirichlet showed flaws in Cauchy's work. This led to Dirichlet's earning the moniker "founder of the theory of Fourier series."

Dirichlet, during his tenure at Göttingen, worked with Dedekind as both teacher and collaborator. At one time a student remarked that "he only knew Dedekind by sight because Dedekind always arrived and left with Dirichlet and was completely eclipsed by him." Dedekind himself wrote that "What is most useful to me is the almost daily association with Dirichlet, with whom I am for the first time beginning to learn properly.... I thank him already for infinitely many things, and no doubt there will be many more."

However, in spite of his accomplishments as a mathematician and mentor, Dirichlet paid little attention to himself. This led one colleague to note, "He is a tall, lanky-looking man, with moustache and beard about to turn grey with a somewhat harsh voice and rather deaf. He was unwashed, with his cup of coffee and cigar. One of his failings is forgetting time, he pulls his watch out, finds it half past three, and runs out without even finishing the sentence."

In addition to his work in mathematics, Dirichlet made some contributions to mathematical physics. In particular, he studied potential theory and the mechanics of systems in equilibrium. Dirichlet was elected a Fellow of the Royal Society in 1855 and is remembered by Crater Dirichlet on the Moon. He died in 1859 at age 54.

**P. ANDREW KARAM**

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