Henri Poincare : biography
Poincare firstly found that thought in an article by Lazarus Immanuel Fuchs, who was the leading expert in the sphere of differential equations in 1880. During the following several years Poincare worked at developing Fuchs’s idea, creating a theory of a new class of functions, which Poincare, with his natural indifference of questions of priority, offered to call Fuchs’s functions or les fonctions fuchsiennes in French. Even in spite of the fact that Poincare had all the reasons to call those functions with his name. The problem was solved when Christian Klein offered to call the functions automorphic. That time remained as a scientific term. Poincare made row of those functions and proved the theorem of addition and the theorem of the ability of uniformization of algebraic curves (that was the twenty-second scientific problem solved by Poincare in 1907). Those discoveries, as contemporaries mentioned, were fairly considered to be the summit of the development of the theory of analytical functions of complex variable in the nineteenth century.
While working at the theory of automorphic functions, Poincare discovered the there was a connection between them and the geometry of Lobachevsky. That point let the scientist explain the questions of those functions with the language of geometry.
After Poincare’s research elliptical functions turned into a limited special case of a more general theory out of the priority field of science.
Differential equations and mathematical physics
After protecting his doctor’s dissertation, which was devoted to studying peculiar points of the system of differential equations, Poincare wrote a series of memoires, called “About curves, determined by differential equations” (since 1881 till 1882 for the equations of the 1 range and since 1885 till 1886 for equations of the 2 grade). In those articles the scientist created a new field of mathematics, which was named qualitative theory of differential equations. Poincare showed that even if a differential equation could not be solved with the known functions, anyway it was possible to get a lot of information about its qualities and ways of behavior of the family of its solving just judging by the kind of an equation. In detail, Poincare studied the character of solving of integration curves on a surface, gave classification of special points (saddle, focus, center and knot), determined the definitions of the utmost circle and the index of circle, proved that the number of the utmost circles was always final, not taking into account a few special cases. Poincare also created a general theory of integral invariations and solutions of equations in the variations. For the equations of finite difference he created a new sphere, which was called asymptonic analysis of solutions. All the achievements he made, he used for his research while working at practical problems of mathematical physics and celestial mechanics, and the methods he used, became the base of his research in topology.
Special points of integral curves:
Poincare spent much time working with differential equations in particular derivative, mainly in his research in the sphere of problems of mathematical physics. He substantially supplied the number of methods of mathematical physics, particularly, he made a major contribution in the theory of potential, the theory of thermal conductivity and studied vibrations of three-dimensional bodies. Works devoted to the topic of explaining Dirichlet’s principal were also made by the scientist. In order to do that he developed a method in his article about equations with particular derivative. The method was called “a sweeping method” or “method de balayage”.
Algebra and theory of numerals
Poincare successfully applied his theoretic-group methods in his first works already. That approach became his most important tool in his further research, beginning with topology and finishing with the relative theory. Poincare was the first to apply the theory of groups to research in the sphere of physics. He was also the first scientist who studied the group of conversion of Lorentz. The scientist also made great contribution to the theory of discrete groups and its representations.