Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's Last Theorem and created analytic number theory. In analysis, he advanced the theory of Fourier series and was one of the first to give the modern formal definition of a function. In mathematical physics, he studied potential theory, boundary-value problems, heat diffusion, and hydrodynamics.

Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular for results named after him.

Biography

Early life (1805–1822)

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. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelette (or more likely ), a small community north east of Liège in Belgium, from which his surname "Lejeune Dirichlet" ("", French for "the youth from Richelette") was derived.

Although his family was not wealthy and he was the youngest of seven children, his parents supported his education. They enrolled him in an elementary school and then private school in hope that he would later become a merchant. The young Dirichlet, who showed a strong interest in mathematics before age 12, persuaded his parents to allow him to continue his studies. In 1817 they sent him to the under the care of Peter Joseph Elvenich, a student his family knew. In 1820, Dirichlet moved to the Jesuit Gymnasium in Cologne, where his lessons with Georg Ohm helped widen his knowledge in mathematics. He left the gymnasium a year later with only a certificate, as his inability to speak fluent Latin prevented him from earning the Abitur.

His first original research, comprising part of a proof of Fermat's Last Theorem for the case , brought him immediate fame, being the first advance in the theorem since Fermat's own proof of the case and Euler's proof for . Adrien-Marie Legendre, one of the referees, soon completed the proof for this case; Dirichlet completed his own proof a short time after Legendre, and a few years later produced a full proof for the case . In June 1825 he was accepted to lecture on his partial proof for the case at the French Academy of Sciences, an exceptional feat for a 20-year-old student with no degree. With the support of Humboldt and Gauss, Dirichlet was offered a teaching position at the University of Breslau. However, as he had not passed a doctoral dissertation, he submitted his memoir on the Fermat theorem as a thesis to the University of Bonn. Again his lack of fluency in Latin rendered him unable to hold the required public disputation of his thesis; after much discussion, the university decided to bypass the problem by awarding him an honorary doctorate in February 1827. Also, the Minister of Education granted him a dispensation for the Latin disputation required for the Habilitation. Dirichlet earned the Habilitation and lectured in the 1827–28 year as a Privatdozent at Breslau. Rebecka was born in Hamburg. In 1816 her parents arranged for her to be baptised at which point she took the names Rebecka Henriette Mendelssohn Bartholdy. She became a part of the notable salon of her parents, Abraham Mendelssohn and his wife Lea, having social contacts with the important musicians, artists and scientists in a highly creative period of German intellectual life. In 1829 she sang a small role in the premiere, given at the Mendelssohn house, of Felix's Singspiel Die Heimkehr aus der Fremde. She later wrote:

<blockquote>My older brother and sister stole my reputation as an artist. In any other family I would have been highly regarded as a musician and perhaps been leader of a group. Next to Felix and Fanny, I could not aspire to any recognition. </blockquote>

In 1832 she married Dirichlet, who was introduced to the Mendelssohn family by Alexander von Humboldt. In 1833 their first son, Walter, was born. She died in Göttingen in 1858.

Berlin (1826–1855)

As soon as he came to Berlin, Dirichlet applied to lecture at the University of Berlin, and the Education Minister approved the transfer and in 1831 assigned him to the faculty of philosophy. The faculty required him to undertake a renewed habilitation qualification, and although Dirichlet wrote a Habilitationsschrift as needed, he postponed giving the mandatory lecture in Latin for another 20 years, until 1851. As he had not completed this formal requirement, he remained attached to the faculty with less than full rights, including restricted emoluments, forcing him to keep in parallel his teaching position at the Military School. In 1832 Dirichlet became a member of the Prussian Academy of Sciences, the youngest member at only 27 years old.

Holding liberal views, Dirichlet and his family supported the 1848 revolution; he even guarded with a rifle the palace of the Prince of Prussia. After the revolution failed, the Military Academy closed temporarily, causing him a large loss of income. When it reopened, the environment became more hostile to him, as officers he was teaching were expected to be loyal to the constituted government. Some of the press who had not sided with the revolution pointed him out, as well as Jacobi and other liberal professors, as "the red contingent of the staff". 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, Dirichlet proved his theorem on arithmetic progressions using concepts from mathematical analysis to tackle an algebraic problem, thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions. In that article, he also noted the difference between the absolute and conditional convergence of series and its impact in what was later called the Riemann series theorem. In 1841, he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers <math>\mathbb{Z}[i]</math>. is still an unsolved problem in number theory despite later contributions by other mathematicians.

Analysis

thumb|right|200px|Dirichlet found and proved the convergence conditions for Fourier series decomposition. Pictured: the first four Fourier series approximations for a [[Square wave (waveform)|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. Before Dirichlet's solution, not only Fourier, but also Poisson and Cauchy had tried unsuccessfully to find a rigorous proof of convergence. The memoir pointed out Cauchy's mistake and introduced Dirichlet's test for the convergence of series. It also introduced the Dirichlet function as an example of a function that is not integrable (the definite integral was still a developing topic at the time) and, in the proof of the theorem for the Fourier series, introduced the Dirichlet kernel and the Dirichlet integral.

Dirichlet also studied the first boundary-value problem, for the Laplace equation, proving the uniqueness of the solution; this type of problem in the theory of partial differential equations was later named the Dirichlet problem after him. A function satisfying a partial differential equation subject to the Dirichlet boundary conditions must have fixed values on the boundary.

Other fields

Dirichlet also worked in mathematical physics, lecturing and publishing research in potential theory (including the Dirichlet problem and Dirichlet principle mentioned above), the theory of heat and hydrodynamics.

Dirichlet also lectured on probability theory and least squares, introducing some original methods and results, in particular for limit theorems and an improvement of Laplace's method of approximation related to the central limit theorem. The Dirichlet distribution and the Dirichlet process, based on the Dirichlet integral, are named after him.

Honors

Dirichlet was elected as a member of several academies:

  • Prussian Academy of Sciences (1832)
  • Saint Petersburg Academy of Sciences (1833) – corresponding member
  • Göttingen Academy of Sciences (1846)
  • French Academy of Sciences (1854) – foreign member
  • Royal Swedish Academy of Sciences (1854)
  • Royal Belgian Academy of Sciences (1855)
  • Royal Society (1855) – foreign member

In 1855 Dirichlet was awarded the civil class medal of the Pour le Mérite order at Alexander von Humboldt's recommendation. The Dirichlet crater on the Moon and the 11665 Dirichlet asteroid are named after him.

Selected publications

A complete bibliography of Dirichlet's published works, including translations thereof and lectures not contained in the Werke, is available in:

References

  • .
  • Johann Peter Gustav Lejeune Dirichlet – Œuvres complètes – Gallica-Math