John Peter Gustav Legenne Dirichlet

On February 13, 1805, the German mathematician Dirichlet was born. He was the founder of analytic number theory and the definer of modern function concepts.

character brief introduction

Dirichlet's full name was Johan Peter Gustav Legenne Dirichlet, a German mathematician who founded the formal definition of modern functions. Dirichlet lived in an era when German mathematics was undergoing a period of gradual transformation from backwardness to prosperity, led by C.F. Gauss. With his outstanding mathematical teaching talents and outstanding achievements in the fields of number theory, analysis and mathematical physics, Dirichlet became a core figure in the German mathematical community after Gauss, who was as famous as C.G.J. Jacobi.

Dirichlet was born in Dylan, and his father was the postmaster. He was educated in Germany and was taught by physicist G. Ohm in high school. After passing the high school graduation exam at the age of 16, his parents wanted him to study law, but Dirichlet had chosen mathematics as his lifelong career. At that time, in the German mathematics community, except for Gauss, who was famous in Europe, generally had a low level; because Gauss did not like teaching, Dirichlet decided to go to the mathematics center to attend university in Paris. He studied in Paris from 1822 to 1826 and was deeply influenced by the mathematician Fourier's work on trigonometric series and mathematical physics. On the other hand, Dirichlet never gave up studying Gauss's classic number theory "Research in Arithmetic" published in 1801. No other mathematician at the time could fully understand Gauss's book, and Dirichlet was the first to truly grasp its essence. It can be said that Gauss and Fourier are the two mathematical seniors who have the greatest influence on Dirichlet's academic studies.

In 1825, Dirichlet submitted his first mathematical paper to the French Academy of Sciences, entitled "The Insolvable of Certain Indeterminate Equations of the Fifth Order." He used algebraic number theory methods to prove Fermat's Last Theorem for n=5; Later, he also proved the case for n=14. In 1826, after returning to China, Dilichlet taught at the University of Breslau, the Berlin Military Academy and the University of Berlin. During his 27 years of teaching and research, due to his clear lectures, profound thoughts, humility and eagerness, he cultivated a group of outstanding mathematicians, which had a huge impact on Germany becoming another international mathematical center in the late 19th century.

In 1831, Dirichlet became a member of the Berlin Academy of Sciences. In 1855, he took over the position of professor at Gedin University by C. F. Gauss. He passed away in the spring of 1859.

main contribution

¡ñ Number theory

Dirichlet's early number theory work in Berlin focused on improving Gauss's proof or expression in Arithmetic Research and other number theory papers. Later, he introduced some more in-depth questions and results into Gauss's theory.

In 1842, Dirichlet began to study forms with Gaussian coefficients and used the "box principle" for the first time, which plays an important role in many arguments in modern number theory. In 1846, he obtained a beautiful and complete result in the article "Complex Unit Theory", which belongs to the unit theory of algebraic number theory, which is now called the Dirichlet Unit Theorem.

In 1863, Dirichlet's "Lectures on Number Theory" was edited and published by his student and friend R. Dedekin. This lecture was not only the final commentary on Gauss's "Research in Arithmetic", but also incorporated his many careful creations in number theory, which were later reprinted many times and became one of the classics of number theory.

¡ñ Analysis

Dirichlet was one of the advocates of strict analysis in the 19th century.

In 1829, he published his most famous article,"On the Convergence of Trigonometric Series." This was the first sufficient condition for the convergence of Fourier series that was strictly proved, and the precise study of trigonometric series theory began.

In 1837, Dirichlet returned to the above topic again and published an article entitled "Representing Completely Arbitrary Functions in Sine and Cosine Series", which expanded the concept of functions commonly used at that time and introduced the modern concept of functions. To illustrate that this rule is completely arbitrary, Dirichlet gave examples of functions with "extremely strange behaviors", which is now called Dirichlet's function. On this basis, Dirichlet established the theory of Fourier series.

¡ñ Mathematical Physics

In 1839, Dirichlet published three mathematical papers related to mechanics, discussing the method of multiple integral estimation, which was used to determine the gravity of an ellipsoid on any particle inside or outside, and began his research on mathematical physics problems. The most important article in this regard was published in 1850, proposing to study the boundary value problem of Laplace's equation (now called the Dirichlet problem or the first boundary value problem). This type of problem is particularly important in thermodynamics and electrodynamics and is a basic topic in the study of mathematical equations.