Augustin-Louis Cauchy

Augustin-Louis Cauchy (21.08.1789-23.05.1857) was a French mathematician, member of. Paris Academy of Sciences (1816), the St. Petersburg Academy of Sciences (1831). Genus. Paris. His first teacher and mentor was his father. K. graduated from Ecole Polytechnique (1807) and the School of Bridges and Roads (1810) in Paris. Some time worked as an engineer means of communication, and in 1813 engaged in scientific pursuits and the teaching. He was appointed a member of the Academy instead of G. Monge. In 1816 memoir K. on the theory of waves on the surface of heavy fluid in the competition of the Paris Academy of Sciences awarded first prize, then K. invited to the Ecole Polytechnique, the Sorbonne and College de France in 1830-1838 to. traveled to Europe, in Paris to. returned in 1838, but because of hostility to the new regime refused to various posts of scientists, not wishing to take the oath until he was offered the chair 'without conditions'.

By Portfolio. belong to different fields of mathematics. There were times when K. every week presented to the Paris Academy of Sciences new memoir. In total, he wrote and published over 800 works on arithmetic and number theory, algebra, calculus, differential equations, theoretical and celestial mechanics, mathematical physics, and t. d. The rapidity with which K. passed from one object to another, partly gave him the opportunity to lay in mathematics many new ways. His 'course analysis' (1821), . 'Summary of lectures on infinitesimal calculus' (1823), . 'Lectures on the applications of analysis to geometry' (1826-1828), . based on the systematic use of the concept of limit, . served as a model for the majority of courses later time,
. In them he gave a definition of continuity of, . explicit construction of the theory of convergent series (in particular, . first to establish the exact conditions for convergence of the Taylor series for this function and has a clear distinction between the convergence of the series in general and the convergence of this function, introduced the concept of radius of convergence, . proved a theorem on the product of two absolutely convergent series and m,
. etc.), gave a definition of the integral as the limit of sums, the proof of the existence of integrals of continuous functions, etc.. Great contribution to. is the fact that he developed the basic theory of analytic functions of complex variable, laid back in 18. L. Euler and M. D 'Alamberom. Especially important are these results K.; geometric representation of a complex variable as a point moves in the plane on one or another path of integration (this idea earlier expressed to. Gauss' etc.), the expression of an analytic function in the form of integral (Cauchy), and hence the expansion of the function in power series, the development of residue theory and its applications to various problems of analysis, etc.

. In the theory of differential equations to
. include: setting one of the most important common problems of the theory of differential equations (Cauchy problem), . basic existence theorems of solutions for the case of real and complex variables (for the latter, he developed the method of majorant method) and the integration of partial differential equations 1-th order (Cauchy's method - the method of characteristic strips),
. In geometry K. generalized the theory of polyhedra, . gave a new way to explore the surface of the 2-order, . investigated touch, . squaring and square curves, . established rules of the application of analysis to geometry, . and the equation of the plane and parametric representation of straight line in space,
. K. proved (1813), that two convex polyhedron with congruent, respectively, and equally spaced edges have equal dihedral angles between corresponding faces. In algebra, it is another to prove the fundamental theorem of the theory of symmetric polynomials, developed the theory of determinants, finding all the main properties, in particular the theorem of multiplication (where K. proceeded from the notion of alternating function). This theorem, he distributed to the matrix. K. belong to the terms 'module' of a complex number, 'involving' complex numbers, etc.. K. Sturm's theorem extended to complex roots. In the theory of numbers K. belong: the proof of Fermat's theorem on multi-coal numbers, . One of the proofs of the reciprocity law, . as well as research on the theory of algebraic integers, . in which he received a number of results, . later in more general terms set by the German mathematician E,
. Kummer. He first studied the general indefinite ternary cubic equation and gave a theorem on indefinite ternary quadratic equations and comparisons with the same module and the general solution. K. also belong study of trigonometry, mechanics, elasticity, optics, astronomy, and t. d. K. was a member of the London Royal Islands, and almost all the Academies of Sciences. Complete Works For. issued by the Paris Academy of Sciences.