CE231 Homework No.2 Junho Song (CE SEMM SID: 14608411)
A Short Biographical Report on
I. His Life
Johann Peter Gustav Lejeune Dirichlet was born in Feb 13, 1805 in Düren, Germany.
He attended the gymnasium in Bonn, and then the Jesuit gymnasium in Cologne. In 1822 he was attracted to Paris by the names of P.S. Laplace, A.M. Legendre, J.Fourier, D.S.Poisson, and A.L.Cauchy. The facilities for a mathematical education there were far better than in Germany, where K.F.Gauss was the only great figure.
He read in Paris Gauss’ Disquisitiones Arithemticce, a work which he never ceased to admire and study. He is said to have slept with Gauss’ Disquisitiones Arithmeticae under his pillow. In any case, he fleshed out many of the concise and rigid proofs. Much in it was simplified by Dirichlet, and thereby placed within easier reach of mathematicians.
His first memoir was presented to the French Academy in 1825 on Fermat’s equation. He became docent in Breslau in 1827. Dirichlet taught at the University of Breslau in 1827 and the University of Berlin from 1828 to 1855.
In 1828 he accepted a position in Berlin, and finally succeeded K.F. Gauss at Göttongen in 1855. Ever since the time of Gauss, Göttongen has been an important center of mathematics. Gauss was not only a great mathematician but he was also interested in the applications of that subject to the solutions of problems of astronomy, physics, and geodetic surveying. As a result of his work, a tradition of contact between mathematics and its applications was established in Göttongen. His successors, the outstanding mathematicians Riemann, Clebsch, and Dirichlet upheld this tradition and usually gave lectures in various branches of theoretical physics as well as in pure mathematics. At the end of the nineteenth and the beginning of the twentieth century, this idea of linking these subjects gained new impetus under the influence of the followers such as Felix Klein.
In 1856-1857 Dirichlet lectured on potential theory in Göttongen. An important contribution to potential theory is due to Dirichlet. In 1876, after his death (He died in May 5, 1859 in Göttingen, Germany), the text of the course was published by F. Grube under the title ‘Vorlesungen über die im umgekehrten Verhältniss des Quadrats der Entfernung wirkenden Kräfte’. Grube had atttened the lectures and in the Preface he says that the book gives an accurate exposition of the subjects treated in the lectures. It is a text book on potential theory, probably one of the first. The subjects are treated from the point of view of a mathematician, but the influences from physics are apparent.
2. His Researches
His main researches can be categorized as follows:
First, Dirichlet is best known for his papers on conditions, for the convergence of trigonometric series and the use of the series to represent arbitrary functions. These series had been used previously by Fourier in solving differential equations. Dirichlet’s acquaintance with J. Fourier led him to investigate Fourier’s series. This work by Dirichlet is published in Crelle's Journal in 1828. Earlier work by Poisson on the convergence of Fourier series was shown to by nonrigorous by Cauchy. Cauchy's work itself was shown to be in error by Dirichlet who wrote of Cauchy's paper "The author of this work himself admits that his proof is defective for certain functions for which the convergence is, however, incontestable." Because of this work Dirichlet is considered the founder of the theory of Fourier series and the conditions for the convergence is called Dirichlet’s Conditions.
Secondly, He produced Dirichlet’s Theorem on prime numbers. In 1826, Dirichlet proved that in any arithmetic progression with first term coprime to the difference there are infinitely many primes. This had been conjectured by Gauss.
Thirdly, in mechanics, he investigated the equilibrium of systems and potential theory. This led him to the Dirichlet problem concerning harmonic functions with given boundary conditions to solve differential equations in the process. He gave his name to the boundary conditions: Dirichlet Boundary Conditions. Gauss and Dirichlet used this method and they obtained with it important results in potential theory. Riemann also used it and he based his theory of complex functions on it. At the turn of the century Hilbert turned his attention to the principle and under certain conditions he proved its validity. The way in which he attacked the problem led to new developments in the calculus of variations, an old branch of mathematics already studied by Euler and Lagrange. These investigations also played a role in the creation of functional analysis, a domain of mathematics that began its development at the end of the 19th century. The origin of the principle is in physics, although this is no longer apparent from the modern formulation. Originally the integral that was to be minimized meant something like the energy of a system.
3. Dirichlet’s Boundary Condition and Others
There are three kinds of boundary conditions for differential equations:
When we specify the value of u on the boundary, we speak of Dirichlet boundary conditions. An example for a vibrating string with its ends at x=0 and x =L, fixed would be
4. References of This Report:
5. Other Sources for Dirichlet: