Search Results for Dirichlet

Biographies

1. Dirichlet biography
   
   • Johann Peter Gustav Lejeune Dirichlet .
     
   • Lejeune Dirichlet's family came from the Belgium town of Richelet where Dirichlet's grandfather lived.
     
   • Many details of the Dirichlet family are given in [Durerner Geschichtsblatter 71 (1982), 31-56.',6)">6] where it is shown that the Dirichlets came from the neighbourhood of Liege in Belgium and not, as many had claimed, from France.
     
   • By the age of 16 Dirichlet had completed his school qualifications and was ready to enter university.
     
   • However, the standards in German universities were not high at this time so Dirichlet decided to study in Paris.
     
   • It is interesting to note that some years later the standards in German universities would become the best in the world and Dirichlet himself would play a hand in the transformation.
     
   • Dirichlet set off for France carrying with him Gauss's Disquisitiones arithmeticae a work he treasured and kept constantly with him as others might do with the Bible.
     
   • In Paris by May 1822, Dirichlet soon contracted smallpox.
     
   • From the summer of 1823 Dirichlet was employed by General Maximilien Sebastien Foy, living in his house in Paris.
     
   • Dirichlet was very well treated by General Foy, he was well paid yet treated like a member of the family.
     
   • In return Dirichlet taught German to General Foy's wife and children.
     
   • Dirichlet's first paper was to bring him instant fame since it concerned the famous Fermat's Last Theorem.
     
   • The cases n = 3 and n = 4 had been proved by Euler and Fermat, and Dirichlet attacked the theorem for n = 5.
     
   • Dirichlet proved case 1 and presented his paper to the Paris Academy in July 1825.
     
   • In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for case 2 which was an extension of his own argument for case 1.
     
   • It is worth noting that Dirichlet made a later contribution proving the n = 14 case (a near miss for the n = 7 case!).
     
   • On 28 November 1825 General Foy died and Dirichlet decided to return to Germany.
     
   • There was a problem for Dirichlet since in order to teach in a German university he needed an habilitation.
     
   • Although Dirichlet could easily submit an habilitation thesis, this was not allowed since he did not hold a doctorate, nor could he speak Latin, a requirement in the early nineteenth century.
     
   • The problem was nicely solved by the University of Cologne giving Dirichlet an honorary doctorate, thus allowing him to submit his habilitation thesis on polynomials with a special class of prime divisors to the University of Breslau.
     
   • There was, however, much controversy over Dirichlet's appointment and the large correspondence between German professors both for and against his appointment is considered in [NTM Schr.
     
   • From 1827 Dirichlet taught at Breslau but Dirichlet encountered the same problem which made him choose Paris for his own education, namely that the standards at the university were low.
     
   • The Military College was not the attraction, of course, rather it was that Dirichlet had an agreement that he would be able to teach at the University of Berlin.
     
   • Dirichlet was appointed to the Berlin Academy in 1831 and an improving salary from the university put him in a position to marry, and he married Rebecca Mendelssohn, one of the composer Felix Mendelssohn's two sisters.
     
   • Dirichlet had a lifelong friend in Jacobi, who taught at Konigsberg, and the two exerted considerable influence on each other in their researches in number theory.
     
   • However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV.
     
   • Dirichlet then made a request for assistance from Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, which was successful.
     
   • Dirichlet obtained leave of absence from Berlin for eighteen months and in the autumn of 1843 set off for Italy with Jacobi and Borchardt.
     
   • Schlafli and Steiner were also with them, Schlafli's main task being to act as their interpreter but he studied mathematics with Dirichlet as his tutor.
     
   • Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845.
     
   • Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties.
     
   • Dirichlet did not accept the offer from Gottingen immediately but used it to try to obtain better conditions in Berlin.
     
   • The quieter life in Gottingen seemed to suit Dirichlet.
     
   • We should now look at Dirichlet's remarkable contributions to mathematics.
     
   • Details are given in [The Mathematical Intelligencer 10 (1988), 13-26.',13)">13] where Rowe discusses the importance of the intellectual and personal relationship between Gauss and Dirichlet.
     
   • Analytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet's memoir of 1837 on the existence of primes in a given arithmetic progression.
     
   • Shortly after publishing this paper Dirichlet published two further papers on analytic number theory, one in 1838 with the next in the following year.
     
   • These papers introduce Dirichlet series and determine, among other things, the formula for the class number for quadratic forms.
     
   • This work led him to the Dirichlet problem concerning harmonic functions with given boundary conditions.
     
   • Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions.
     
   • Dirichlet's work is published in Crelle's Journal in 1828.
     
   • Cauchy's work itself was shown to be in error by Dirichlet who wrote of Cauchy's paper:- .
     
   • Because of this work Dirichlet is considered the founder of the theory of Fourier series.
     
   • Riemann, who was a student of Dirichlet, wrote in the introduction to his habilitation thesis on Fourier series that it was Dirichlet [Mathematics in Berlin (Berlin, 1998), 33-40.',11)">11]:- .
     
   • In [Dictionary of Scientific Biography (New York 1970-1990).',1)">1] Dirichlet's character and teaching qualities are summed up as follows:- .
     
   • At age 45 Dirichlet was described by Thomas Hirst as follows:- .
     
   • Koch, in [Mathematics in Berlin (Berlin, 1998), 33-40.',11)">11], sums up Dirichlet's contribution writing that:- .
     
   • important parts of mathematics were influenced by Dirichlet.
     
   • With Dirichlet began the golden age of mathematics in Berlin.
     
   • Honours awarded to Lejeune Dirichlet .
     
   • Lunar featuresCrater Dirichlet .
     
   • http://www-history.mcs.st-andrews.ac.uk/Biographies/Dirichlet.html .
     

2. Riemann biography
   
   • Riemann moved from Gottingen to Berlin University in the spring of 1847 to study under Steiner, Jacobi, Dirichlet and Eisenstein.
     
   • The main person to influence Riemann at this time, however, was Dirichlet.
     
   • Riemann was bound to Dirichlet by the strong inner sympathy of a like mode of thought.
     
   • Dirichlet loved to make things clear to himself in an intuitive substrate; along with this he would give acute, logical analyses of foundational questions and would avoid long computations as much as possible.
     
   • His manner suited Riemann, who adopted it and worked according to Dirichlet's methods.
     
   • In proving some of the results in his thesis Riemann used a variational principle which he was later to call the Dirichlet Principle since he had learnt it from Dirichlet's lectures in Berlin.
     
   • The Dirichlet Principle did not originate with Dirichlet, however, as Gauss, Green and Thomson had all made use if it.
     
   • Gauss's chair at Gottingen was filled by Dirichlet in 1855.
     
   • The Dirichlet Principle which Riemann had used in his doctoral thesis was used by him again for the results of this 1857 paper.
     
   • Weierstrass, however, showed that there was a problem with the Dirichlet Principle.
     
   • He fully recognised the justice and correctness of Weierstrass's critique, but he said, as Weierstrass once told me, that he appealed to Dirichlet's Principle only as a convenient tool that was right at hand, and that his existence theorems are still correct.
     
   • We return at the end of this article to indicate how the problem of the use of Dirichlet's Principle in Riemann's work was sorted out.
     
   • In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Gottingen on 30 July.
     
   • Finally let us return to Weierstrass's criticism of Riemann's use of the Dirichlet's Principle.
     
   • Weierstrass had shown that a minimising function was not guaranteed by the Dirichlet Principle.
     
   • Weierstrass firmly believed Riemann's results, despite his own discovery of the problem with the Dirichlet Principle.
     
   • He asked his student Hermann Schwarz to try to find other proofs of Riemann's existence theorems which did not use the Dirichlet Principle.
     
   • In 1901 Hilbert mended Riemann's approach by giving the correct form of Dirichlet's Principle needed to make Riemann's proofs rigorous.
     

3. Dedekind biography
   
   • Gauss died in 1855 and Dirichlet was appointed to fill the vacant chair at Gottingen.
     
   • This was an extremely important event for Dedekind who found working with Dirichlet extremely profitable.
     
   • He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations.
     
   • Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two.
     
   • recalled in later years that he only knew Dedekind by sight because Dedekind always arrived and left with Dirichlet and was completely eclipsed by him.
     
   • 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; he is always completely amiable towards me, and he tells me without beating about the bush what gaps I need to fill and at the same time he gives me the instructions and the means to do it.
     
   • Dirichlet supported his application writing that Dedekind was 'an exceptional pedagogue'.
     
   • Among Dedekind's other notable contributions to mathematics were his editions of the collected works of Peter Dirichlet, Carl Gauss, and Georg Riemann.
     
   • Dedekind's study of Dirichlet's work did, in fact, lead to his own study of algebraic number fields, as well as to his introduction of ideals.
     
   • Dedekind edited Dirichlet's lectures on number theory and published these as Vorlesungen uber Zahlentheorie in 1863.
     
   • Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death.
     

4. Christoffel biography
   
   • Elwin Christoffel was noted for his work in mathematical analysis, in which he was a follower of Dirichlet and Riemann.
     
   • Christoffel studied at the University of Berlin from 1850 where he was taught by Borchardt, Eisenstein, Joachimsthal, Steiner and Dirichlet.
     
   • It was Dirichlet who had the greatest influence on him and Christoffel is rightly thought of as a student of Dirichlet's.
     
   • He returned to Montjoie where his mother was in poor health but read widely from the works of Dirichlet, Riemann and Cauchy.
     
   • Between 1865 and 1871 Christoffel published four important papers on potential theory, three of them dealing with the Dirichlet problem.
     
   • In our opinion Christoffel's teacher Dirichlet, belongs to the next most important group of mathematicians which includes (in chronological order of birth) Jacobi, Kummer, Kronecker, Dedekind, Cantor and Klein.
     

5. Kellogg biography
   
   • He attended lectures by Hilbert who suggested he undertake research on the Dirichlet problem for plane regions bounded by a finite number of plane curves which met at points where the boundary was not differentiable.
     
   • Fredholm had just published a major work on the Dirichlet problem but Fredholm's methods did not apply to the regions which Hilbert suggested Kellogg investigate.
     
   • In January of the following year he received his doctorate for his dissertation Zur Theorie der Integralgleichungen und des Dirichlet'schen Prinzips on the Dirichlet problem but he remained in Germany to the end of the academic year, returning to the United States to take up the post of instructor in mathematics at Princeton.
     
   • In 1908 he published three papers, namely Potential functions on the boundary of their regions of definition and Double distributions and the Dirichlet problem, both in the Transactions of the American Mathematical Society, and A necessary condition that all the roots of an algebraic equation be real in the Annals of Mathematics.
     

6. Kummer biography
   
   • This led to Jacobi, and later Dirichlet, corresponding with Kummer on mathematical topics and they soon realised the great potential for the highest level of mathematics that Kummer possessed.
     
   • In 1839, although still a school teacher, Kummer was elected to the Berlin Academy on Dirichlet's recommendation.
     
   • In 1840 Kummer married a cousin of Dirichlet's wife.
     
   • In 1842, with strong support from Jacobi and Dirichlet, he was appointed a full professor at the University of Breslau, now Wroclaw in Poland.
     
   • In 1855 Dirichlet left Berlin to succeed Gauss at Gottingen.
     

7. Trudinger biography
   
   • First there is the paper On the Dirichlet problem for quasilinear uniformly elliptic equations in n variables in which he extended previous work by his supervisor David Gilbarg, Olga Ladyzhenskaya and others on the solvability of the classical Dirichlet problem in bounded domains for certain second order quasilinear uniformly elliptic equations.
     
   • Secondly, in the paper The Dirichlet problem for nonuniformly elliptic equation he exploited the maximum principle to formulate general conditions for solvability of the Dirichlet problem for certain nonlinear elliptic equations.
     
   • The authors restrict themselves mainly to the theory of the Dirichlet problem.
     

8. Kronecker biography
   
   • Kronecker became a student at Berlin University in 1841 and there he studied under Dirichlet and Steiner.
     
   • Back in Berlin he worked on his doctoral thesis on algebraic number theory under Dirichlet's supervision.
     
   • Dirichlet commented on the thesis saying that in it Kronecker showed:- .
     
   • In 1855 Kummer came to Berlin to fill the vacancy which occurred when Dirichlet left for Gottingen.
     

9. Heine biography
   
   • After three semesters at Gottingen, Heine returned to the University of Berlin where he where he was taught by Dirichlet.
     
   • The thesis was dedicated to Dirichlet.
     
   • The first proof of this theorem was given by Dirichlet in his lectures of 1862 (published 1904) before Heine proved it in 1872.
     
   • Dugac shows that Dirichlet used the idea of a covering and a finite subcovering more explicitly than Heine.
     

10. Jacobi biography
    
    • However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV.
      
    • Let us now return to Dirichlet and Alexander von Humboldt's attempts to help obtain support for Jacobi's trip to Italy.
      
    • Dirichlet's request to Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, was successful and Jacobi received a grant to allow him to spend time in Italy.
      
    • He set off for Italy with Borchardt and Dirichlet and, after stopping in several towns and attending a mathematical meeting in Lucca, they arrived in Rome on 16 November 1843.
      

11. Eisenstein biography
    
    • By the time he was seventeen, although he was still at school, he began to attend lectures by Dirichlet and other mathematicians at the University of Berlin.
      
    • In 1842 he bought a French translation of Gauss's Disquisitiones arithmeticae and, like Dirichlet, he became fascinated by the number theory which he read there.
      
    • Early in 1852, at Dirichlet's request, Eisenstein was elected to the Berlin Academy.
      

12. Beurling biography
    
    • Beurling worked on the theory of generalized functions, differential equations, harmonic analysis, Dirichlet series and potential theory.
      
    • The concepts of energy and the Dirichlet integral took Beurling to a global axiomatic theory called the theory of Dirichlet spaces for complex functions.
      

13. Mandelbrojt biography
    
    • Continuing to work at the Rice Institute during World War II, Mandelbrojt continued to publish important work In 1944 he published a series of lectures he had given at the Institute under the title Dirichlet series:- .
      
    • This monograph consists of a series of lectures delivered by the author and is not a complete treatment of general Dirichlet series.
      
    • In 1969 Mandelbrojt published Series de Dirichlet.
      

14. Fredholm biography
    
    • Already in 1895 after a seminar lecture in 1895 he had talked about Dirichlet's problem as one of elimination.
      
    • In fact much of this work was accomplished during the months of 1899 which Fredholm spent in Paris studying the Dirichlet problem with Poincare, Emile Picard, and Hadamard.
      
    • In 1900 a preliminary report on his theory of Fredholm integral equations was published as Sur une nouvelle methode pour la resolution du probleme de Dirichlet.
      

15. Haselgrove biography
    
    • at Manchester he published papers on Dirichlet functions and the Riemann hypothesis, ray paths in the ionosphere, numerical integration using quasi-random numbers, two-point boundary-value problems, and some geometrical puzzles.
      
    • His tables of Dirichlet L-functions were deposited with the Royal Society but not published although he did publish (with D Davies) the paper The evaluation of Dirichlet L-functions (1961) describing the methods used.
      

16. Stoilow biography
    
    • The second volume has the following chapter headings: The Dirichlet problem; Local properties of harmonic functions; The Dirichlet problem for multiply-connected domains; The Dirichlet integral and the minimum principle; Green's function, Lindelof's principle, the principle of harmonic measure; Harmonic measure; Riemann surfaces; Analytic functions on closed Riemann surfaces; Analytic functions on open Riemann surfaces; Regularly and normally exhaustible Riemann surfaces.
      

17. Schwarz biography
    
    • Although Riemann had given a proof of the theorem that any simply connected region of the plane can be mapped conformally onto a disc, his proof involved using the Dirichlet problem.
      
    • Weierstrass had shown that Dirichlet's solution to this was not rigorous, see [Rend.
      
    • Schwarz also gave the alternating method for solving the Dirichlet problem which soon became a standard technique.
      

18. Mazya biography
    
    • Solution of Dirichlet's problem for an equation of elliptic type (Russian) was published in 1959 and Classes of domains and imbedding theorems for function spaces (Russian) in 1960.
      
    • The Dirichlet problem for an arbitrary order elliptic equation in a domain with a cut off tubular neighbourhood of a smooth closed submanifold is considered in the second chapter.
      
    • The main term of an asymptotic expansion for a solution of the Dirichlet problem for the Laplacian in a three-dimensional domain with a narrow slit is obtained in the third chapter.
      

19. Joachimsthal biography
    
    • In 1836 he entered the University of Berlin where he was taught by Dirichlet and Steiner.
      
    • His colleagues included many famous mathematicians who all contributed to his development of mathematical ideas, in particular Eisenstein, Dirichlet, Jacobi, Steiner and Borchardt.
      
    • Influenced by the work of Jacobi, Dirichlet and Steiner, Joachimsthal wrote on the theory of surfaces where he made substantial contributions, particularly to the problem of normals to conic sections and second degree surfaces.
      

20. Schlafli biography
    
    • Later that year Steiner, Jacobi and Dirichlet travelled to Rome and took Schlafli with them as an interpreter.
      
    • Schlafli gained greatly from discussions with these leading mathematicians, in particular having a daily lesson in number theory from Dirichlet.
      

21. Chebyshev biography
    
    • In Berlin he met Dirichlet:- .
      
    • It was of great interest for me to become acquainted with the celebrated geometer Lejeune-Dirichlet.
      

22. Selberg biography
    
    • Subsequent events are not entirely clear but Selberg published two papers An elementary proof of the prime number theorem and An elementary proof of Dirichlet's theorem about primes in an arithmetic progression in volume 50 of the Annals of Mathematics.
      
    • automorphic functions, Dirichlet series.
      

23. Gronwall biography
    
    • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.
      

24. Seidel biography
    
    • Seidel entered the University of Berlin in 1840 and studied under Dirichlet and Encke.
      
    • In the autumn of 1843 Jacobi left Konigsberg on the grounds of ill health and set off for Italy with Borchardt, Dirichlet, Schlafli and Steiner.
      

25. Johnson biography
    
    • How do Bayesians justify using conjugate priors on grounds other than mathematical convenience? In the 1920s the Cambridge philosopher William Ernest Johnson in effect characterized symmetric Dirichlet priors for multinomial sampling in terms of a natural and easily assessed subjective condition.
      
    • Johnson's proof can be generalized to include asymmetric Dirichlet priors and those finitely exchangeable sequences with linear posterior expectation of success.
      

26. Renyi biography
    
    • The first result, regarding the representation of an even number, is an approximation to the unproved Goldbach conjecture and supersedes an earlier proof of the same proposition by Estermann (1932) which made use of an unproved generalized Riemann hypothesis for all Dirichlet L-series.
      
    • Other papers published early in his career include: On a Tauberian theorem of O Szasz (1946); Integral formulae in the theory of convex curves (1947); On the minimal number of terms of the square of a polynomial (1947); On some new applications of the method of Academician I M Vinogradov (1947); (with Yu V Linnik) On certain hypotheses in the theory of Dirichlet characters (Russian) (1947).
      

27. Meissel biography
    
    • He also had contacts with Dirichlet who was teaching at the University of Berlin at the time Meissel was studying there.
      
    • Meissel must be judged as a classical mathematician, continuing a tradition from an earlier epoch associated with names like Euler, Laplace, Legendre, Gauss, Jacobi, and Dirichlet.
      

28. Suetuna biography
    
    • In particular he read Hardy and Littlewood's paper The approximate functional equation in the theory of the zeta function with applications to the divisor problems of Dirichlet and Piltz which appeared in the Proceedings of the London Mathematical Society.
      
    • This book, based mainly on the Riemann zeta-functions and L-functions, is a unique exposition of the analytical theory of numbers in a modern sense as can be seen from the chapter headings: I) Riemann's zeta-functions; II) Hecke's L-functions; III) Dirichlet's L-functions; and IV) Artin's L-series.
      

29. Riesz Marcel biography
    
    • The first period of his work, from the beginning of his doctoral research up to around the beginning of World War I, concentrated on the theory of series, in particular the summability theory of power series, trigonometric series and Dirichlet series.
      
    • In a joint work with Hardy The general theory of Dirichlet's series, published by Cambridge University Press in 1915, he introduced Riesz means.
      

30. Carlson biography
    
    • Such names as Carlson inequality, Carlson - Levin constants, Carlson theorem in complex analysis, Polya - Carlson theorem on rational functions and Carlson theorem on Dirichlet series are well-known in mathematics (see [Inequalities (Cambridge, 1934).
      
    • Carlson, in a series of papers, investigated Dirichlet series and proved in 1922 that if f (z) = sigman1inf ann-z is convergent in Re z ≥ 0 and bounded in every Re z > δ > 0, then, for each > 0, .
      

31. Bohr Harald biography
    
    • Harald Bohr worked on Dirichlet series, and applied analysis to the theory of numbers.
      
    • Bohr's interest in which functions could be represented by a Dirichlet series led him to devise the theory of almost periodic functions.
      

32. Courant biography
    
    • His thesis was entitled Uber die Anwendung des Dirichletschen Prinzipes auf die Probleme der konformen Abbildung (On the application of Dirichlet's principle to the problems of conformal mappings).
      
    • For his habilitation thesis Courant again worked on the Dirichlet principle.
      

33. Borchardt biography
    
    • He studied at Berlin from 1836 under Dirichlet then, in 1839, he went to Konigsberg and studied under Bessel, Franz Neumann and Jacobi.
      
    • Dirichlet and Steiner were also in Rome at the same time and it proved a useful time for Borchardt.
      

34. Chudakov biography
    
    • Chudakov is also famed as the author of the classic monograph Introduction to the theory of Dirichlet L-functions (1947) which was widely used by number theory experts.
      
    • The text assumes only that the reader is familiar with the elements of number theory and complex variable theory, and goes on to develop the theory of characters and Dirichlet L-functions.
      

35. Bjerknes Carl biography
    
    • Dirichlet lectured to Carl Bjerknes in Gottingen on hydrodynamics and Bjerknes became so interested in that topic that he spent the rest of his life researching in that area.
      
    • He began his investigations from a result that Dirichlet had proved in the lectures he attended, namely that a ball can move at a constant speed without the action of external forces through a frictionless fluid.
      

36. Bachmann biography
    
    • In 1856 he went from Berlin to Gottingen so that he could continue to study courses by Dirichlet who had just left Berlin to succeed to Gauss's chair in Gottingen.
      
    • There he worked on number theory and he was awarded his habilitation in 1864 for a thesis on complex units which had been a topic which he had been inspired to work on though the lectures by Dirichlet which he had attended.
      

37. Gauss biography
    
    • Dirichlet's principle was mentioned without proof.
      
    • From the mathematical community only Jacobi and Dirichlet were present, but Gauss received many messages and honours.
      

38. Steklov biography
    
    • He reduced problems of this type to boundary-value problems of Dirichlet type using rigorous mathematical analysis.
      
    • In addition to the work for his master's thesis and his doctoral thesis referred to above, he reduced problems to boundary value problems of Dirichlet type where Laplace's equation must be solved on a surface.
      

39. Plessner biography
    
    • In 1921 Plessner went to Gottingen where he took courses on Dirichlet series and Galois theory by Edmund Landau; algebraic number fields by Emmy Noether; and calculus of variations by Courant.
      

40. Weber biography
    

41. Minkowski biography
    
    • Already at this stage in his education he was reading the work of Dedekind, Dirichlet and Gauss.
      

42. Du Bois-Reymond biography
    
    • The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function.
      

43. Hilbert biography
    
    • Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more.
      

44. De Bruijn biography
    
    • Also in 1943, in addition to his doctoral thesis, he published On the absolute convergence of Dirichlet series, On the number of solutions of the system ..
      

45. Mellin biography
    
    • He applied this technique systematically in a long series of papers to the study of the gamma function, hypergeometric functions, Dirichlet series, the Riemann zeta function and related number-theoretic functions.
      

46. Chowla biography
    
    • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).
      

47. Atkinson biography
    
    • In fact much of his early research followed on from this beginning with papers such as A summation formula for p(n), the partition function (1939), The mean value of the zeta-function on the critical line (1941), A divisor problem (1941), The Abel summation of certain Dirichlet series (1948), A mean value property of the Riemann zeta-function (1948), The mean-values of arithmetical functions (1949), and The mean-value of the Riemann zeta function (1949).
      

48. Wright biography
    
    • These topics are: prime numbers; congruences and the quadratic reciprocity law; continued fractions; irrational, algebraic and transcendental numbers; quadratic fields; arithmetical functions, their order of magnitude and the Dirichlet or power series which generate them; partitions and representations of numbers as sums of squares, cubes and higher powers; Diophantine approximation; and the geometry of numbers.
      

49. Rajagopal biography
    
    • In several papers he studied the relation between the growth of the mean values of an entire function and that of its Dirichlet series.
      

50. Oleinik biography
    
    • Oleinik considers equations satisying Dirichlet boundary conditions and ones which satisfy Neumann boundary conditions.
      

51. Hille biography
    
    • Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series.
      

52. Neumann Carl biography
    
    • During the 1860s Neumann wrote papers on the Dirichlet principle and the 'logarithmic potential', a term he coined.
      

53. Bosanquet biography
    
    • He also wrote on the convergence and summability of Dirichlet series and studied specific kinds of summability such as summability factors for Cesaro means.
      

54. Hecke biography
    
    • Schoeneberg describes Hecke's contributions to a number of topics which he lists as follows: Hilbert modular functions, Dedekind zeta functions, arithmetical notions and methods, elliptic modular forms of level N, algebraic functions, Dirichlet series with functional equation, Hecke-operators Tn, and physics where he made contributions to the kinetic theory of gases.
      

55. Lebesgue biography
    
    • He also made major contributions in other areas of mathematics, including topology, potential theory, the Dirichlet problem, the calculus of variations, set theory, the theory of surface area and dimension theory.
      

56. Fenyo biography
    
    • This covers topics such as the Green function technique for a one-dimensional Sturm-Liouville problem, and Dirichlet and Carl Neumann problems for the two- and three-dimensional Laplacian operator.
      

57. Cramer Harald biography
    
    • Also influenced by G H Hardy, Cramer's research resulted in the award of a PhD in 1917 for his thesis On a class of Dirichlet series.
      

58. Liouville biography
    
    • Another blow to Liouville was the death of Dirichlet in 1859.
      

59. Korkin biography
    
    • He had read, and with his wonderful memory could then recall, most works by Abel, Dirichlet, Euler, Fourier, Gauss, Jacobi, Lagrange, Laplace, Legendre, Monge, and Poisson.
      

60. Mathews biography
    
    • In his two volume work Theory of numbers (1892) topics covered included Gauss's theory of quadratic forms and their development by mathematicians such as Dirichlet, Eisenstein and Smith.
      

61. Schroeter biography
    
    • At Berlin he was taught by Dirichlet and Steiner.
      

62. Landau biography
    
    • He submitted this habilitation thesis in 1901, only two years after his doctorate, consisted of his work on Dirichlet series, a topic in analytic number theory.
      

63. Linnik biography
    
    • The large sieve method led Linnik to study Dirichlet's L-functions.
      

64. Lipschitz biography
    
    • Following the custom of that time to study at different universities, Lipschitz went from Konigsberg to Berlin where he studied under Dirichlet.
      

65. Blaschke biography
    
    • This work in particular shows how he was developing ideas due to Steiner who had worked on this topic but was subsequently criticised by Dirichlet for not giving existence proofs.
      

66. Morera biography
    
    • He developed the study of the harmonic functions, applying results due to Pizzetti, finding a simple expression for the inner and outer gravitational field of an ellipsoid, solving the Dirichlet problem.
      

67. Hirst biography
    
    • In particular he attended lectures by Dirichlet and Steiner, being strongly influenced by Steiner to undertake further research on geometry.
      

68. Cournot biography
    
    • Cournot remained in Paris and, along with his fellow student Dirichlet, was taught mathematics at the Sorbonne by Lacroix and Hachette.
      

69. Severi biography
    
    • His most impressive work came before he went to Rome but, despite spending less time on mathematics, after this he still managed to produce work of the greatest importance like the solution of the Dirichlet problem and his development of the theory of rational equivalence.
      

70. Weingarten biography
    
    • At the University of Berlin Weingarten attended lectures on potential theory given by Dirichlet.
      

71. Mitchell biography
    
    • The method was not completely reliable, although it did well on problems with homogeneous Dirichlet boundary conditions on at least two sides of the square.
      

72. Wolf biography
    
    • He moved to Vienna in 1836, studying there for two years before going to Berlin in 1838 where he attended lectures by Encke, Dirichlet, Poggendorf, Steiner and Crelle.
      

73. Bernstein Sergi biography
    
    • He studied for his Master's degree at Kharkov, continuing his way through Hilbert's Problems by solving the Twentieth on the analytic solution of Dirichlet's problem for a wide class of non-linear elliptic equations.
      

74. Nikodym biography
    
    • the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory and the foundations of quantum mechanics.
      

75. Dini biography
    
    • Dini looked at infinite series and generalised results such as a theorem of Kummer and one of Riemann, the ideas for which had first emerged in work of Dirichlet.
      

76. Bremermann biography
    
    • In 1959 he used ideas on his earlier work to give a generalisation of the Dirichlet problem.
      

77. Feldman biography
    
    • They are based on Dirichlet's Schubfachprinzip and on two interpolation formulae ..
      

78. Petersson biography
    
    • So what is now completely standard linear algebra for all mathematicians was, when Petersson published Konstruktion der samtlichen Losungen einer Riemannschen Funktionalgleichung durch Dirichlet-Reihen mit Eulerscher Produktentwicklung in 1939, at the forefront of the latest research.
      

79. Steinhaus biography
    
    • He was awarded his doctorate, with distinction, for a dissertation Neue Anwendungen des Dirichlet'schen Prinzips in 1911.
      

80. Cantor Moritz biography
    
    • He then studied at Berlin with Dirichlet and Steiner.
      

81. Gateaux biography
    
    • I think that the generalization of the Dirichlet problem should be more difficult.
      

82. Crelle biography
    
    • In addition to Abel, mathematicians such as Dirichlet, Eisenstein, Grassmann, Hesse, Jacobi, Kummer, Lobachevsky, Mobius, Plucker, von Staudt, Steiner, and Weierstrass all had their early works made famous by publication in Crelle's journal.
      

83. Roch biography
    
    • Schlomilch received his university education at Berlin under Dirichlet and Steiner.
      

84. Dynkin biography
    
    • a class of measure-valued Markov processes [which] can be used to give probabilistic solutions to certain nonlinear PDE's in a way which is analogous to the classical solution of the Dirichlet problem by means of Brownian motion.
      

85. Mertens biography
    
    • Many people are aware of Mertens contributions since his elementary proof of the Dirichlet theorem appears in most modern textbooks.
      

86. Zaremba biography
    
    • He studied elliptic equations and in particular contributed to the Dirichlet principle.
      

87. Legendre biography
    
    • Of course today we attribute the law of quadratic reciprocity to Gauss and the theorem concerning primes in an arithmetic progression to Dirichlet.
      

88. Aronhold biography
    
    • Aronhold went with Jacobi to Berlin where he continued his studies under Dirichlet and Steiner.
      

89. Zygmund biography
    
    • The general part of the book is followed by three chapters, one on entire and meromorphic functions, one on elliptic functions, and one on G(s), Z(s) and Dirichlet series.
      

90. Vinogradov biography
    
    • He generalised results of Voronoy on the Dirichlet divisor problem which allowed him to obtain estimates for the number of integral points between a given curve y = f (x) and the x-axis.
      

91. Chebotaryov biography
    
    • The density theorem generalised Dirichlet's theorem on primes in an arithmetical progression giving a method used by Artin in 1927 in his reciprocity law, a result considered the main result of class field theory.
      

92. Artin biography
    
    • He defined a new type of L-series, which generalised Dirichlet's L-series, yet was quite different in nature.
      

93. Cantor biography
    
    • This was a difficult problem which had been unsuccessfully attacked by many mathematicians, including Heine himself as well as Dirichlet, Lipschitz and Riemann.
      

94. Kober biography
    
    • To give an illustration of his work, we note that he published four papers in 1940: On Dirichlet's singular integral; On some generalisations of Laguerre polynomials; On fractional integrals and derivatives; and Some remarks on Hankel transforms (with Arthur Erdelyi).
      

95. Poincare biography
    
    • He began his contributions to this topic in 1883 with a paper in which he used the Dirichlet principle to prove that a meromorphic function of two complex variables is a quotient of two entire functions.
      

96. Suvorov biography
    
    • One of the many innovations in Suvorov's work was new methods which he introduced to help in the understanding of metric properties of mappings with bounded Dirichlet integral.
      
    • In 1981 Suvorov published The metric theory of prime ends and boundary properties of plane mappings with bounded Dirichlet integrals (Russian).
      
    • Some applications of this theory to the study of mappings with bounded Dirichlet integral are also described.
      
    • Finally, they give some applications in potential theory (Dirichlet problem).
      
    • This final monograph continues to develop the topics considered in his two earlier monographs Families of plane topological mappings (1965) and The metric theory of prime ends and boundary properties of plane mappings with bounded Dirichlet integrals (1981).
      

97. Wiener Norbert biography
    
    • From 1923 he investigated Dirichlet's problem, producing work which had a major influence on potential theory.
      

98. Voronoy biography
    
    • He then looked at contributions by Dirichlet and Hermite, showing that none of the above provided a satisfactory generalisation.
      

99. Weierstrass biography
    
    • A letter from Dirichlet to the Prussian Minister of Culture written in 1855 strongly supported Weierstrass being given a university appointment.
      

History Topics

1. Fermat's last theorem
   
   • Case 2(i) was proved by Dirichlet and presented to the Paris Academie des Sciences in July 1825.
     Go directly to this paragraph
     
   • In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).
     Go directly to this paragraph
     
   • In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14.
     Go directly to this paragraph
     
   • It showed why Dirichlet had so much difficulty, for although Dirichlet's n = 14 proof used similar (but computationally much harder) arguments to the earlier cases, Lame had to introduce some completely new methods.
     Go directly to this paragraph
     
   • By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat's Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbers B2 , B4 , ..
     Go directly to this paragraph
     

2. function concept
   
   • Fourier's work would lead eventually to the clarification of the function concept when in 1829 Dirichlet proved results concerning the convergence of Fourier series, thus clarifying the distinction between a function and its representation.
     
   • Dirichlet, in 1837, accepted Fourier's definition of a function and immediately after giving this definition he defined a continuous function (using continuous in the modern sense).
     
   • Dirichlet also gave an example of a function defined on the interval [ 0, 1] which is discontinuous at every point, namely f(x) which is defined to be 0 if x is rational and 1 if x is irrational.
     
   • Certainly Dirichlet's everywhere discontinuous function will not be a function under Lobachevsky's definition.
     
   • One person defines functions essentially in Euler's sense, the other requires that y must change with x according to a law, without giving an explanation of this obscure concept, the third defines it in Dirichlet's manner, the fourth does not define it at all.
     

3. Hirst's diary
   
   • Dirichlet .
     
   • Lejeune Dirichlet: .
     
   • I thought, as we sat each at the end of the sofa, and the smoke of our cigars carried question and answer to and fro, and intermingled in graceful curves before it rose to the ceiling and mixed with the common atmospheric air, "If all be well, we will smoke our friendly cigar together many a time yet, good-natured Lejeune Dirichlet".
     
   • (31 Oct 1852) Dirichlet cannot be surpassed for richness of material and clear insight into it: as a speaker he has no advantages - there is nothing like fluency about him, and yet an eye and understanding make it indispensable: without an effort you would not notice his hesitating speech.
     
   • (20 Feb 1853) Dirichlet has ..
     

4. function concept references
   
   • J Lutzen, The development of the concept of function from Euler to Dirichlet.
     
   • F A Medvedev, Lobacevskii's and Dirichlet's definition of the concept of function (Russian), Istor.-Mat.
     

5. Ring Theory
   
   • n = 5Legendre and Dirichlet1825 .
     
   • n = 14Dirichlet1832 .
     

6. function concept references
   
   • J Lutzen, The development of the concept of function from Euler to Dirichlet.
     
   • F A Medvedev, Lobacevskii's and Dirichlet's definition of the concept of function (Russian), Istor.-Mat.
     

7. Cartography references
   
   • N S Ermolaeva, Mathematical cartography and D A Grave's method for solving the Dirichlet problem (Russian), Istor.-Mat.
     

8. Prime numbers
   
   • (Dirichlet proved that every arithmetic progression : {a + bn | n belongs N} with a, b coprime contains infinitely many primes.) .
     Go directly to this paragraph
     

9. Cartography references
   
   • N S Ermolaeva, Mathematical cartography and D A Grave's method for solving the Dirichlet problem (Russian), Istor.-Mat.
     

Famous Curves

Societies etc

1. Young Mathematician prize
   
   • for works on Poisson-Dirichlet measures.
     

2. Lunar features
   

3. Lunar features
   
   • (W) (L) Dirichlet .
     

4. Lunar features
   
   • Dirichlet .
     

5. Fellow of the Royal Society
   
   • Gustav L Dirichlet 1855 .
     

References

1. References for Dirichlet
   
   • References for Lejeune Dirichlet .
     
   • E E Kummer, Peter Gustav Lejeune Dirichlet, in L Kronecker and L Fuchs, G Lejeune Dirichlets Werke (Berlin, 1889-97).
     
   • P L Butzer, Dirichlet and his role in the founding of mathematical physics, Arch.
     
   • Todestag des Mathematikers Johann Peter Gustav Lejeune Dirichlet (1805-1859), Mitbegrunder der mathematischen Physik im deutschsprachigen Raum, Sudhoffs Arch.
     
   • P L Butzer, M Jansen and H Zilles, Johann Peter Gustav Lejeune Dirichlet (1805-1859): Genealogie und Werdegang, Durerner Geschichtsblatter 71 (1982), 31-56.
     
   • H Davenport, Dirichlet, Math.
     
   • H Fischer, Dirichlet's Contribution to Mathematical Probability Theory, Historia Mathematica 21 (1994), 39-63.
     
   • H J Koch, P G Lejeune Dirichlet zu seinem 175.
     
   • H Koch, Gustav Peter Lejeune Dirichlet, in Mathematics in Berlin (Berlin, 1998), 33-40.
     
   • Kh Kokh, On the occasion of the 175th anniversary of the birth of P G Lejeune Dirichlet (Russian), Istor.-Mat.
     
   • D E Rowe, Gauss, Dirichlet and the Law of Biquadratic Reciprocity, The Mathematical Intelligencer 10 (1988), 13-26.
     
   • G Schubring, The three parts of the Dirichlet 'Nachlass', Historia Math.
     
   • G Schubring, Die Promotion von P G Lejeune Dirichlet : Biographische Mitteilungen zum Werdegang Dirichlets, NTM Schr.
     
   • A Shields, Lejeune Dirichlet and the birth of analytic number theory : 1837-1839, The Mathematical Intelligencer 11 (1989), 7-11.
     
   • http://www-history.mcs.st-andrews.ac.uk/References/Dirichlet.html .
     

2. References for Jacobi
   
   • P G Lejeune Dirichlet, Gedachtnissrede auf Carl Gustav Jacob Jacobi, in Nachrufe auf Berliner Mathematiker des 19.
     

3. References for Weierstrass
   
   • K-R Biermann, Dirichlet uber Weierstrass, Praxis Math.
     

4. References for Heine
   
   • P Dugac, Sur la correspondance de Borel et le theoreme de Dirichlet- Heine- Weierstrass- Borel- Schoenflies- Lebesgue, Arch.
     

5. References for Gateaux
   
   • Jacques Hadamard, Sur le principe de Dirichlet, Bull.
     

6. References for Bachmann
   
   • O Neumann, Vorlesungsnachschriften zu Dirichlet-Dedekind-Kummerschen Themen : verfasst von Paul Bachmann (1837-1920), Natur, Mathematik und Geschichte.
     

7. References for Gauss
   
   • D E Rowe, Gauss, Dirichlet and the Law of Biquadratic Reciprocity, The Mathematical Intelligencer 10 (1988), 13-26.
     

8. References for Grave
   
   • N S Ermolaeva, Mathematical cartography and D A Grave's method for solving the Dirichlet problem (Russian), Istor.-Mat.
     

Additional material

1. Thomas Bromwich: 'Infinite Series
   
   • The use of AbeI's and Dirichlet's names for the tests given in Art.
     
   • To illustrate the general theory, a short discussion of Dirichlet's integrals and of the Gamma integrals is given; it is hoped that these proofs will be found both simple and rigorous.
     

2. Goursat: 'Cours d'analyse mathématique
   
   • - Probleme de Dirichlet.
     
   • Probleme de Dirichlet dans l'espace.
     

3. Mathematicians and Music
   
   • He referred to the cultures of mathematics and music "not merely as having arithmetic for their common parent but as similar in their habits and affections." "May not Music be described," he wrote, "as the Mathematic of Sense, Mathematic as the Music of reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music, - Music the dream, Mathematic the working life, - each to receive its consummation from the other when the human intelligence, elevated to the perfect type, shall shine forth glorified in some future Mozart-Dirichlet, or Beethoven-Gauss - a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz"! .
     
   • Dirichlet seemed to be sensible to the charms of music in a similar manner.
     

4. EMS obituary
   
   • 1", and the two led me to read about ideals in the old book, Dirichlet's edition of Dedekind's "Theory of Numbers", about a third of which is taken up with the Theory of Ideals.
     

5. Mathematicians and Music 3
   
   • It is sometimes asserted that the first mathematical proof of Fourier's results, with the limits of arbitrariness of the function carefully stated, was given by Dirichlet in his classic memoirs of 1829 and 1837.
     

6. Donald C Spencer's publications
   
   • D C Spencer, Dirichlet's principle on manifolds, Studies in Math.
     

7. Henry Baker addresses the British Association in 1913
   
   • And, although the principle of Thomson and Dirichlet, which relates to the potential problem referred to, was expounded by Gauss, and accepted by Riemann, and remains to-day in our standard treatise on Natural Philosophy, there can be no doubt that, in the form in which it was originally stated, it proves just nothing.
     

8. G H Hardy addresses the British Association in 1922, Part 2
   
   • The first step towards a solution was made by Dirichlet, who proved for the first time, in 1837, that any such arithmetical progression contains an infinity of primes.
     

9. Publications of Eduard Heine
   
   • E Heine, uber einige Voraussetzungen beim Beweise des Dirichlet'schen Principes, Clebsch Ann.
     

10. Konrad Knopp: Texts
    
    • 5.5 Abel's and Dirichlet's tests and their generalizations .
      

11. Carathéodory: 'Conformal representation
    
    • Nevertheless, about fifty years after Riemann, Hilbert was able to prove rigorously that the particular problem which arose in Riemann's work does possess a solution; this theorem is known as Dirichlet's Principle.
      

12. Poincaré on intuition in mathematics
    
    • I shall take as second example Dirichlet's principle on which rest so many theorems of mathematical physics; to-day we establish it by reasonings very rigorous but very long; heretofore, on the contrary, we were content with a very summary proof.
      

13. Three Sadleirian Professors
    
    • One has already been mentioned; the others are The Integration of Functions of a Single Variable (1905) and The General Theory of Dirichlet's Series (1915, in collaboration with M Riesz).
      

14. Publications of Eduard Heine
    
    • E Heine, uber einige Voraussetzungen beim Beweise des Dirichlet'schen Principes, Clebsch Ann.
      

15. Henry Baker addresses the British Association in 1913, Part 2
    
    • But in the land of Kummer and Gauss and Dirichlet the subject to-day claims the allegiance of many eager minds.
      

16. Publications of Albert Wangerin
    
    • Laplace, Ivory, Gauss, Chasles und Dirichlet: Uber die Anziehung homogener Ellipsoide (W Engelmann, Leipzig, 1890).
      

17. G H Hardy addresses the British Association in 1922
    
    • The first step towards a solution was made by Dirichlet, who proved for the first time, in 1837, that any such arithmetical progression contains an infinity of primes.
      

Quotations

1. Quotations by Jacobi
   
   • Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous mathematical proof is.
     
   • When Gauss says that he has proved something, it is very clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain ..
     

2. A quotation by Tietze
   
   • The story was told that the young Dirichlet had as a constant companion all his travels, like a devout man with his prayer book, an old, worn copy of the Disquisitiones Arithmeticae of Gauss.
     

3. Quotations by Gauss
   
   • The total number of Dirichlet's publications is not large: jewels are not weighed on a grocery scale.
     

Chronology

1. Mathematical Chronology
   
   • Dirichlet gives a general definition of a function.
     
   • The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more.
     
   • Fredholm develops his theory of integral equations in Sur une nouvelle methode pour la resolution du probleme de Dirichlet.
     

2. Chronology for 1890 to 1900
   
   • The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more.
     
   • Fredholm develops his theory of integral equations in Sur une nouvelle methode pour la resolution du probleme de Dirichlet.
     

3. Chronology for 1900 to 1910
   
   • The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more.
     
   • Fredholm develops his theory of integral equations in Sur une nouvelle methode pour la resolution du probleme de Dirichlet.
     

4. Chronology for 1830 to 1840
   
   • Dirichlet gives a general definition of a function.