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“Dirichlet开始了柏林数学的黄金时代”
这是Koch对Dirichlet的评价。
Dirichlet(1805——1859),曾经因为没有德国(博士)学位,从法国“海归”后无法在大学教书。他20岁已经证明了费马定理(n=5时),得到一个荣誉学位才得到了在Breslau大学的任职资格。不到四年,他就发现这个大学的水平并不适合他,就在23岁(1828年)去了柏林。他在柏林教了27年书,包括在军事学院和柏林大学。
50岁时,他接替了Gauss在哥廷根大学的职位,并且要求离开军事学院,但这个学院似乎不乐意。他终究还是离开了那里,直到四年后因病逝世于哥廷根。
Dirichlet是物理大地测量中经常提到的一个名字。Dirichlet边值问题,催生了一些高质量的论文。
除了以上提到的Gauss以外,Fourier,Laplace,Legendre,Poisson都是物理大地测量教材中常见的名字。
以下是Dirichlet的五张照片以及延伸阅读材料:
延伸阅读:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Dirichlet.html
Lejeune Dirichlet's family came from the
His father was the postmaster of Düren, the town of his birth situated about halfway between
... an unusually attentive and well-behaved pupil who was particularly interested in history as well as mathematics.
After two years at the Gymnasium in
Dirichlet set off for
From the summer of 1823 Dirichlet was employed by General Maximilien Sébastien Foy, living in his house in
Dirichlet's first paper was to bring him instant fame since it concerned the famous Fermat's Last Theorem. The theorem claimed that for n > 2 there are no non-zero integers x, y, z such that xn + yn = zn. The cases n = 3 and n = 4 had been proved by Euler and Fermat, and Dirichlet attacked the theorem for n = 5. Now if n = 5 then one of x, y, z is even and one is divisible by 5. There are two cases: case 1 is when the number divisible by 5 is even, while case 2 is when the even number and the one divisible by 5 are distinct. Dirichlet proved case 1 and presented his paper to the Paris Academy in July 1825. Legendre was appointed one of the referees and he was able to prove case 2 thus completing the proof for n = 5. The complete proof was published in September 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
From 1827 Dirichlet taught at Breslau but Dirichlet encountered the same problem which made him choose
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 Königsberg, and the two exerted considerable influence on each other in their researches in number theory.
In the 1843 Jacobi became unwell and diabetes was diagnosed. He was advised by his doctor to spend time in
Dirichlet did not remain in
Dirichlet did not accept the offer from Göttingen immediately but used it to try to obtain better conditions in
The quieter life in Göttingen seemed to suit Dirichlet. He had more time for research and some outstanding research students. However, sadly he was not to enjoy the new life for long. In the summer of 1858 he lectured at a conference in Montreux but while in the Swiss town he suffered a heart attack. He returned to Göttingen, with the greatest difficulty, and while gravely ill had the added sadness that his wife died of a stroke.
We should now look at Dirichlet's remarkable contributions to mathematics. We have already commented on his contributions to Fermat's Last Theorem made in 1825. Around this time he also published a paper inspired by Gauss's work on the law of biquadratic reciprocity. Details are given in [13] where Rowe discusses the importance of the intellectual and personal relationship between Gauss and Dirichlet.
He proved in 1837 that in any arithmetic progression with first term coprime to the difference there are infinitely many primes. This had been conjectured by Gauss. Davenport wrote in 1980 .
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.
His work on units in algebraic number theory Vorlesungen über Zahlentheorie (published 1863) contains important work on ideals. He also proposed in 1837 the modern definition of a function:-
If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.
In mechanics he investigated the equilibrium of systems and potential theory. These investigations began in 1839 with papers which gave methods to evaluate multiple integrals and he applied this to the problem of the gravitational attraction of an ellipsoid on points both inside and outside. He turned to Laplace's problem of proving the stability of the solar system and produced an analysis which avoided the problem of using series expansion with quadratic and higher terms disregarded. This work led him to the Dirichlet problem concerning harmonic functions with given boundary conditions. Some work on mechanics later in his career is of quite outstanding importance. In 1852 he studied the problem of a sphere placed in an incompressible fluid, in the course of this investigation becoming the first person to integrate the hydrodynamic equations exactly.
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. These series had been used previously by Fourier in solving differential equations. Dirichlet's work is published in Crelle's Journal in 1828. Earlier work by Poisson on the convergence of Fourier series was shown to be non-rigorous 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. Riemann, who was a student of Dirichlet, wrote in the introduction to his habilitation thesis on Fourier series that it was Dirichlet [11]:-
... who wrote the first profound paper about the subject.
In [1] Dirichlet's character and teaching qualities are summed up as follows:-
He was an excellent teacher, always expressing himself with great clarity. His manner was modest; in his later years he was shy and at times reserved. He seldom spoke at meetings and was reluctant to make public appearances.
At age 45 Dirichlet was described by Thomas Hirst as follows:-
He is a rather tall, lanky-looking man, with moustache and beard about to turn grey with a somewhat harsh voice and rather deaf. He was unwashed, with his cup of coffee and cigar. One of his failings is forgetting time, he pulls his watch out, finds it past three, and runs out without even finishing the sentence.
Koch, in [11], sums up Dirichlet's contribution writing that:-
... important parts of mathematics were influenced by Dirichlet. His proofs characteristically started with surprisingly simple observations, followed by extremely sharp analysis of the remaining problem. With Dirichlet began the golden age of mathematics in
Article by: J J O'Connor and E F Robertson
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