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===Breslau, Preussi (1825–28)===
General Foy kuoli marraskuussa 1825 ja menetettyään työpaikkansa Dirichlet'n täytyi palata Preussiin. Fourier ja Poisson esittelivät hänet [[Alexander von Humboldt|Humboldtille]], Humboldt tuki Dirichlet'tä muun muassa kirjoittamalla suosituskirjeen Gaussille ylistäen Dirichlet'n lahjakkuutta Fermat'n lauseen käsittelyssä.<ref name=Goldstein>{{cite book| last = Goldstein| first = Cathérine | coauthors = Catherine Goldstein, Norbert Schappacher, Joachim Schwermer| title=The shaping of arithmetic: after C.F. Gauss's Disquisitiones Arithmeticae| year=2007| publisher=Springer| location = | isbn= 978-3-540-20441-1| pages= 204–208}}</ref> WithHumboldtin theja supportGaussin of Humboldt and Gauss,tuella Dirichlet waspääsi offeredopetustöihin aja teachingpystyi positionjatkamaan at the [[University of Wrocław|University of Breslau]] (now the University of Wrocław in [[Poland]])työskentelyään. 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 a [[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.<ref name=Elstrodt/>
== Julkaisuja ==
While in Breslau, Dirichlet continued his number theoretic research, publishing important contributions to the [[Quartic reciprocity|biquadratic reciprocity]] law which at the time was a focal point of Gauss' research. Alexander von Humboldt took advantage of these new results, which had also drawn enthusiastic praise from [[Friedrich 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 only able to get him a trial position at the [[Prussian Military Academy]] in Berlin while remaining nominally employed by the University of Breslau. The probation was extended for three years until the position becoming definitive in 1831.
===Berlin (1826–1855)===
[[image:Rebecka Mendelssohn - Zeichnung von Wilhelm Hensel 1823.jpg|left|thumb|230px|Dirichlet was married from 1832 to [[Rebecka Mendelssohn]]. They had two children, Walter (born 1833) and Flora (born 1845). Drawing by [[Wilhelm Hensel]], 1823]]
After moving to Berlin, Humboldt introduced Dirichlet to the [[Salon (gathering)|great salon]]s held by the banker [[Abraham Mendelssohn Bartholdy]] and his family. Their house was a weekly gathering point of the Berlin artists and scientists, including Abraham's children [[Felix Mendelssohn Bartholdy]] and [[Fanny Mendelssohn]], both outstanding musicians, and the painter [[Wilhelm Hensel]] (Fanny's husband). Dirichlet showed great interest in Abraham's daughter [[Rebecka Mendelssohn]], whom he married in 1832. In 1833 their first son, Walter, was born.
As soon as he came to Berlin, Dirichlet applied to lecture at the [[Humboldt University of Berlin|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.<ref name=Elstrodt/>
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 and mathematical physics. He advised the doctoral theses of several important German mathematicians, as [[Gotthold Eisenstein]], [[Leopold Kronecker]], [[Rudolf Lipschitz]] and [[Carl Wilhelm Borchardt]], while being influential in the mathematical formation of many other scientists, including [[Elwin Bruno Christoffel]], [[Wilhelm Eduard Weber|Wilhelm Weber]], [[Eduard Heine]], [[Philipp Ludwig von Seidel|Ludwig von Seidel]] and [[Julius Weingarten]]. At the Military Academy Dirichlet managed to introduce [[differential calculus|differential]] and [[integral calculus]] in the curriculum, significantly raising the level of scientific education there. However, in time he started feeling that his double teaching load, at the Military academy and at the University, started weighing down on the time available for his research.<ref name=Elstrodt/>
While in Berlin, Dirichlet kept in contact with other mathematicians. In 1829, during a trip, he met [[Carl Gustav Jacob Jacobi|Jacobi]], at the time professor of mathematics at [[Königsberg 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 met [[Joseph 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 by [[Ernst Kummer]], at the time a school teacher. 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.
In 1843, when Jacobi fell ill, Dirichlet traveled to Königsberg to help him, then obtained for him the assistance of [[Frederick William IV of Prussia|King Friedrich Wilhelm IV]]'s personal physician. When the medic recommended Jacobi to spend some time in Italy, he joined him on the trip together with his family. They were accompanied to Italy by [[Ludwig Schläfli]], who came as a translator; as he was strongly interested in mathematics, during the trip both Dirichlet and Jacobi lectured him, later Schläfli becoming an important mathematician himself.<ref name=Elstrodt/> 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 the [[Heidelberg University]] tried to recruit Dirichlet, Jacobi provided von Humboldt the needed support in order to obtain a doubling of Dirichlet's pay at the University in order to keep him in Berlin; however, even now he wasn't paid a full professor wage and he could not leave the Military Academy.<ref name=Calinger>{{cite book| last = Calinger| first = Ronald| title=Vita mathematica: historical research and integration with teaching| year=1996| publisher=Cambridge University Press| location = | isbn= 978-0-88385-097-8| pages= 156–159}}</ref>
Holding liberal views, Dirichlet and his family supported the [[Revolutions of 1848 in the German states|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 to whom he was teaching would ordinarily be expected to be loyal to the constituted government. A portion of the press who were not with the revolution pointed him out, as well as Jacobi and other liberal professors, as "the red contingent of the staff".<ref name=Elstrodt/>
In 1849 Dirichlet participated, together with his friend Jacobi, to the jubilee of Gauss' doctorate.
===Göttingen (1855–59)===
Despite Dirichlet's expertise and the honours he received, and although by 1851 he had finally completed all formal requirements for a full professor, the issue of raising his payment at the University still dragged and he still couldn't leave the Military Academy. In 1855, upon Gauss' death, the [[University 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 follow him as a mathematics professor in Berlin.<ref name=James/>
Dirichlet enjoyed his time in Göttingen as the lighter teaching load allowed him more time for research and, also, he got in close contact with the new generation of researchers, especially [[Richard Dedekind]] and [[Bernhard Riemann]]. After moving to Göttingen he was able to obtain a small annual payment for Riemann in order to retain him in the teaching staff there. Dedekind, Riemann, [[Moritz Cantor]] and [[Alfred Enneper]], although they had all already earned their PhDs, attended Dirichlet's classes to study with him. Dedekind, who felt that there were significant gaps at the time in his mathematics education, considered that the occasion to study with Dirichlet made him "a new human being".<ref name=Elstrodt/> He later edited and published Dirichlet's lectures and other results in [[number theory]] under the title {{lang|de|''[[Vorlesungen über Zahlentheorie]]''}} (''Lectures on Number Theory'').
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.<ref name=James/> Dirichlet's brain is preserved in the department of physiology at the University of Göttingen, along with the brain of Gauss. The Academy in Berlin honored him with a formal memorial speech held by Kummer in 1860, and later ordered the publication of his collected works edited by Kronecker and [[Lazarus Fuchs]].
== Mathematics research ==
{{Further|List of topics named after Gustav Lejeune Dirichlet}}
===Number theory===
[[Number theory]] was Dirichlet's main research interest,<ref name=Princeton>{{cite book| last = Gowers| first = Timothy | coauthors = June Barrow-Green, Imre Leader| title=The Princeton companion to mathematics| year=2008| publisher=Princeton University Press| location = | isbn= 978-0-691-11880-2| pages= 764–765}}</ref> 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 character]]s and [[Dirichlet L-function|L-functions]].<ref name=Princeton/><ref name=Kanemitsu>{{cite book| last = Kanemitsu| first = Shigeru| coauthors = Chaohua Jia| title=Number theoretic methods: future trends | year=2002| publisher=Springer| location = | isbn= 978-1-4020-1080-4| pages= 271–274}}</ref> Also, in the article he noted the difference between the [[Absolute convergence|absolute]] and [[conditional convergence]] of [[Series (mathematics)|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 (mathematics)|ring]] of [[Gaussian integer]]s <math>\mathbb{Z}[i]</math>.<ref name=Elstrodt/>
In a couple of papers in 1838 and 1839 he proved the first [[class number formula]], for [[quadratic form]]s (later refined by his student Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general [[number field]]s.<ref name=Elstrodt/> Based on his research of the structure of the [[unit group]] of [[quadratic field]]s, he proved the [[Dirichlet unit theorem]], a fundamental result in [[algebraic number theory]].<ref name=Kanemitsu/>
He first used the [[pigeonhole principle]], a basic counting argument, in the proof of a theorem in [[diophantine approximation]], later named after him [[Dirichlet's approximation theorem]]. He published important contributions to [[Fermat's last theorem]], for which he proved the cases ''n'' = 5 and ''n'' = 14, and to the [[quartic reciprocity|biquadratic reciprocity law]].<ref name=Elstrodt/> The [[Dirichlet divisor problem]], for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.
=== Analysis ===
[[Image:Fourier Series.svg|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]].]]
Inspired by the work of his mentor in Paris, Dirichlet published in 1829 a famous memoir giving the [[Dirichlet conditions|conditions]], showing for which functions the convergence of the [[Fourier series]] holds. Before Dirichlet's solution, not only Fourier, but also Poisson and [[Augustin-Louis Cauchy|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 that not any function is 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]].<ref name=Bressoud>{{cite book| last = Bressoud| first = David M.| title=A radical approach to real analysis | year=2007| publisher=MAA| location = | isbn= 978-0-88385-747-2| pages= 218–227}}</ref>
Dirichlet also studied the first [[boundary value problem]], for the [[Laplace equation]], proving the unicity of the solution; this type of problem in the theory of [[partial differential equation]]s was later named the [[Dirichlet problem]] after him.<ref name=Princeton/> In the proof he notably used the principle that the solution is the function that minimizes the so-called [[Dirichlet's energy|Dirichlet energy]]. Riemann later named this approach the [[Dirichlet principle]], although he knew it had also been used by Gauss and by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]].<ref name=Elstrodt/>
===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 (mathematics)|function]] by the property that "to any x there corresponds a single finite y", but then restricts his attention to [[piecewise continuous]] functions. Based on this, he is credited with introducing the modern concept for a function, as opposed to the older vague understanding of a function as an analytic formula.<ref name=Elstrodt/> [[Imre Lakatos]] cites [[Hermann 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".<ref name=Lakatos>{{cite book| last = Lakatos| first = Imre| title=Proofs and refutations: the logic of mathematical discovery| year=1976| publisher=Cambridge University Press| location = | isbn= 978-0-521-29038-8| pages= 151–152}}</ref>
===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]].<ref name=Princeton/> He improved on [[Lagrange]]'s work on [[conservative system]]s by showing that the condition for [[Mechanical equilibrium|equilibrium]] is that the [[potential energy]] is minimal.<ref name=Leine>{{cite book| last = Leine| first = Remco|coauthors = Nathan van de Wouw| title=Stability and convergence of mechanical systems with unilateral constraints| year=2008| publisher=Springer| location = | isbn= 978-3-540-76974-3| pages= 6}}</ref>
Although he didn't publish much in the field, Dirichlet lectured on [[probability theory]] and [[least squares]], introducing some original methods and results, in particular for [[Asymptotic theory (statistics)|limit theorems]] and an improvement of [[Laplace's method]] of approximation related to the [[central limit theorem]].<ref name=Fischer>{{cite journal | last = Fischer| first = Hans| journal = Historia Mathematica |volume=21 |issue=1 |pages=39–63 |publisher=Elsevier | title = Dirichlet's contributions to mathematical probability theory | work = |date = February 1994| url = http://www.sciencedirect.com/science/article/pii/S031508608471007X | format = | doi = | accessdate = 2011-10-18}}</ref> The [[Dirichlet distribution]] and the [[Dirichlet process]], based on the Dirichlet integral, are named after him.
==Honours==
Dirichlet was elected as a member of several academies:<ref name=Royal>{{cite journal | last = | first = | journal = Proceedings of the Royal Society of London|volume=10 |issue= |pages=xxxviii–xxxix|publisher=Taylor and Francis | title = Obituary notices of deceased fellows| work = | year = 1860| url = | format = | doi = | accessdate = }}</ref>
* [[Prussian Academy of Sciences]] (1832)
* [[Russian Academy of Sciences|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)
* [[The Royal Academies for Science and the Arts of Belgium|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 von Humboldt's recommendation. The [[Dirichlet (crater)|Dirichlet crater]] on the [[Moon]] and the [[11665 Dirichlet]] asteroid are named after him.
== Selected publications ==
* {{cite book
| last = Lejeune Dirichlet
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