Gustav Lejeune Dirichlet was born on 13 February 1805 inDüren, a town on the left bank of theRhine which at the time was part of theFirst French Empire, reverting toPrussia after theCongress 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 likelyRichelle [fr] ), a small community 5 km (3 miles) north east ofLiège inBelgium, from which his surname "Lejeune Dirichlet" ("le jeune de Richelette",French for "the youth from Richelette") was derived.[3]
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 theGymnasium Bonn [de] under the care ofPeter Joseph Elvenich, a student his family knew. In 1820, Dirichlet moved to theJesuit Gymnasium inCologne, where his lessons withGeorg Ohm helped widen his knowledge in mathematics. He left the gymnasium a year later with only a certificate, as his inability to speak fluentLatin prevented him from earning theAbitur.[3]
Dirichlet again persuaded his parents to provide further financial support for his studies in mathematics, against their wish for a career in law. As Germany provided little opportunity to study higher mathematics at the time, with onlyGauss at theUniversity of Göttingen who was nominally a professor ofastronomy and anyway disliked teaching, Dirichlet decided to go toParis in May 1822. There he attended classes at theCollège de France and at theUniversity of Paris, learning mathematics fromHachette among others, while undertaking private study of Gauss'sDisquisitiones Arithmeticae, a book he kept close for his entire life. In 1823 he was recommended toGeneral Maximilien Foy, who hired him as a private tutor to teach his childrenGerman, the wage finally allowing Dirichlet to become independent from his parents' financial support.[4]
His first original research, comprising part of a proof ofFermat's Last Theorem for the casen = 5, brought him immediate fame, being the first advance in the theorem sinceFermat's own proof of the casen = 4 andEuler's proof forn = 3.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 casen = 14.[5] In June 1825 he was accepted to lecture on his partial proof for the casen = 5 at theFrench Academy of Sciences, an exceptional feat for a 20-year-old student with no degree.[3] His lecture at the Academy had also put Dirichlet in close contact withFourier andPoisson, who raised his interest intheoretical physics, especially Fourier'sanalytic theory of heat.
As General Foy died in November 1825 and he could not find any paying position in France, Dirichlet had to return to Prussia. Fourier and Poisson introduced him toAlexander von Humboldt, who had been called to join the court of KingFriedrich Wilhelm III. Humboldt, planning to makeBerlin a centre of science and research, immediately offered his help to Dirichlet, sending letters in his favour to the Prussian government and to thePrussian Academy of Sciences. Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an unusual amount of praise that "Dirichlet showed excellent talent".[6] With the support of Humboldt and Gauss, Dirichlet was offered a teaching position at theUniversity of Breslau. However, as he had not passed a doctoral dissertation, he submitted his memoir on the Fermat theorem as a thesis to theUniversity 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 anhonorary doctorate in February 1827. Also, the Minister of Education granted him a dispensation for the Latin disputation required for theHabilitation. Dirichlet earned the Habilitation and lectured in the 1827–28 year as aPrivatdozent atBreslau.[3]
While in Breslau, Dirichlet continued his number-theoretic research, publishing important contributions to thebiquadratic reciprocity law which at the time was a focal point of Gauss's research. Alexander von Humboldt took advantage of these new results, which had also drawn enthusiastic praise fromFriedrich Bessel, to arrange for him the desired transfer to Berlin. Given Dirichlet's young age (he was 23 years old at the time), Humboldt was able to get him only a trial position at thePrussian Military Academy in Berlin while remaining nominally employed by the University of Breslau. The probation was extended for three years until the position becoming definite in 1831.
Dirichlet was married in 1832 to Rebecka Mendelssohn. They had two children, Walter (born 1833) and Flora (born 1845). Drawing byWilhelm Hensel, 1823
After Dirichlet's move to Berlin, Humboldt introduced him to thegreat salons held by the bankerAbraham Mendelssohn Bartholdy and his family. Their house was a weekly gathering point for Berlin artists and scientists, including Abraham's childrenFelix andFanny Mendelssohn, both outstanding musicians, and the painterWilhelm Hensel (Fanny's husband). Dirichlet showed great interest in Abraham's other daughter Rebecka, whom he married in 1832.
Rebecka Henriette Lejeune Dirichlet (née Rebecka Mendelssohn; 11 April 1811 – 1 December 1858) was a granddaughter ofMoses Mendelssohn and the youngest sister ofFelix Mendelssohn andFanny Mendelssohn.[7][8] Rebecka was born inHamburg.[9] In 1816 her parents arranged for her to bebaptised at which point she took the names Rebecka Henriette Mendelssohn Bartholdy.[10] She became a part of the notablesalon 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'sSingspielDie Heimkehr aus der Fremde. She later wrote:
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.[11]
As soon as he came to Berlin, Dirichlet applied to lecture at theUniversity of Berlin, and the Education Minister approved the transfer and in 1831 assigned him to the faculty ofphilosophy. The faculty required him to undertake a renewedhabilitation qualification, and although Dirichlet wrote aHabilitationsschrift 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 thePrussian Academy of Sciences, the youngest member at only 27 years old.[3]
Dirichlet had a good reputation with students for the clarity of his explanations and enjoyed teaching, especially as his University lectures tended to be on the more advanced topics in which he was doing research: number theory (he was the first German professor to give lectures on number theory), analysis andmathematical physics. He advised the doctoral theses of several important German mathematicians, asGotthold Eisenstein,Leopold Kronecker,Rudolf Lipschitz andCarl Wilhelm Borchardt, while being influential in the mathematical formation of many other scientists, includingElwin Bruno Christoffel,Wilhelm Weber,Eduard Heine,Ludwig von Seidel andJulius Weingarten. At the Military Academy, Dirichlet managed to introducedifferential andintegral calculus in the curriculum, raising the level of scientific education there. However, he gradually started feeling that his double teaching load, at the Military academy and at the university, was limiting the time available for his research.[3]
While in Berlin, Dirichlet kept in contact with other mathematicians. In 1829, during a trip, he metCarl Jacobi, at the time professor of mathematics atKönigsberg University. Over the years they kept meeting and corresponding on research matters, in time becoming close friends. In 1839, during a visit to Paris, Dirichlet metJoseph Liouville, the two mathematicians becoming friends, keeping in contact and even visiting each other with the families a few years later. In 1839, Jacobi sent Dirichlet a paper byErnst Kummer, at the time a schoolteacher. Realizing Kummer's potential, they helped him get elected in the Berlin Academy and, in 1842, obtained for him a full professor position at the University of Breslau. In 1840 Kummer married Ottilie Mendelssohn, a cousin of Rebecka's.
In 1843, when Jacobi fell ill, Dirichlet traveled to Königsberg to help him, then obtained for him the assistance ofKing Friedrich Wilhelm IV's personal physician. When the physician recommended that Jacobi spend some time in Italy, Dirichlet joined him on the trip together with his family. They were accompanied to Italy byLudwig Schläfli, who came as a translator; as he was strongly interested in mathematics, both Dirichlet and Jacobi lectured to him during the trip, and he later became an important mathematician himself.[3] The Dirichlet family extended their stay in Italy to 1845, their daughter Flora being born there. In 1844, Jacobi moved to Berlin as a royal pensioner, their friendship becoming even closer. In 1846, when theHeidelberg University tried to recruit Dirichlet, Jacobi provided von Humboldt the needed support to obtain a doubling of Dirichlet's pay at the university in order to keep him in Berlin; however, even then he was not paid a full professor wage and could not leave the Military Academy.[13]
Holding liberal views, Dirichlet and his family supported the1848 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".[3]
In 1849 Dirichlet participated, together with his friend Jacobi, in the jubilee of Gauss's doctorate.
Despite Dirichlet's expertise and the honours he received, and even though, by 1851, he had finally completed all formal requirements for a full professor, the issue of raising his pay at the university still dragged on and he was still unable to leave the Military Academy. In 1855, upon Gauss's death, theUniversity of Göttingen decided to call Dirichlet as his successor. Given the difficulties faced in Berlin, he decided to accept the offer and immediately moved to Göttingen with his family.Kummer was called to assume his position as a professor of mathematics in Berlin.[4]
Dirichlet enjoyed his time in Göttingen, as the lighter teaching load allowed him more time for research and he came into close contact with the new generation of researchers, especiallyRichard Dedekind andBernhard Riemann. After moving to Göttingen he was able to obtain a small annual stipend for Riemann to retain him in the teaching staff there. Dedekind, Riemann,Moritz Cantor andAlfred Enneper, although they had all already earned their PhDs, attended Dirichlet's classes to study with him. Dedekind, who felt that there were gaps in his mathematics education, considered that the occasion to study with Dirichlet made him "a new human being".[3] He later edited and published Dirichlet's lectures and other results innumber theory under the titleVorlesungen über Zahlentheorie (Lectures on Number Theory).
In the summer of 1858, during a trip toMontreux, Dirichlet suffered a heart attack. On 5 May 1859, he died in Göttingen, several months after the death of his wife Rebecka.[4] 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 andLazarus Fuchs.
Dirichlet found and proved the convergence conditions for Fourier series decomposition. Pictured: the first four Fourier series approximations for asquare wave.
Inspired by the work of his mentor in Paris, Dirichlet published in 1829 a famous memoir giving theconditions, showing for which functions the convergence of theFourier series holds.[17] Before Dirichlet's solution, not only Fourier, but also Poisson andCauchy had tried unsuccessfully to find a rigorous proof of convergence. The memoir pointed out Cauchy's mistake and introducedDirichlet's test for the convergence of series. It also introduced theDirichlet function as an example of a function that is not integrable (thedefinite integral was still a developing topic at the time) and, in the proof of the theorem for the Fourier series, introduced theDirichlet kernel and theDirichlet integral.[18]
Dirichlet also studied the firstboundary-value problem, for theLaplace equation, proving the uniqueness of the solution; this type of problem in the theory ofpartial differential equations was later named theDirichlet problem after him. A function satisfying a partial differential equation subject to the Dirichlet boundary conditions must have fixed values on the boundary.[14] In the proof he notably used the principle that the solution is the function that minimizes the so-calledDirichlet energy. Riemann later named this approach theDirichlet principle, although he knew it had also been used by Gauss and byLord Kelvin.[3]
While trying to gauge the range of functions for which convergence of the Fourier series can be shown, Dirichlet defines afunction by the property that "to anyx there corresponds a single finitey", but then restricts his attention topiecewise continuous functions. Based on this, he is credited with introducing the modern concept of a function, as opposed to the older vague understanding of a function as an analytic formula.[3]Imre Lakatos citesHermann Hankel as the early origin of this attribution, but disputes the claim saying that "there is ample evidence that he had no idea of this concept [...] for instance when he discusses piecewise continuous functions, he says that at points of discontinuity, the function has two values".[19]
^Dudenredaktion (2015).Duden – Das Aussprachewörterbuch: Betonung und Aussprache von über 132.000 Wörtern und Namen [Duden – The Pronouncing Dictionary: accent and pronunciation of more than 132.000 words and names]. Duden - Deutsche Sprache in 12 Bänden (in German). Vol. 6. 312.ISBN978-3-411-91151-6.
^Krantz, Steven (2011).The Proof is in the Pudding: The Changing Nature of Mathematical Proof. Springer. pp. 55–58.ISBN978-0-387-48908-7.
^Goldstein, Cathérine; Catherine Goldstein; Norbert Schappacher; Joachim Schwermer (2007).The shaping of arithmetic: after C.F. Gauss's Disquisitiones Arithmeticae. Springer. pp. 204–208.ISBN978-3-540-20441-1.
^Calinger, Ronald (1996).Vita mathematica: historical research and integration with teaching. Cambridge University Press. pp. 156–159.ISBN978-0-88385-097-8.
^Leine, Remco; Nathan van de Wouw (2008).Stability and convergence of mechanical systems with unilateral constraints. Springer. p. 6.ISBN978-3-540-76974-3.
^"Obituary notices of deceased fellows".Proceedings of the Royal Society of London.10. Taylor and Francis:xxxviii–xxxix. 1860.doi:10.1098/rspl.1859.0002.S2CID186209363.
