Augustin Louis Cauchy Демьянов В.П. О.Л.Коши – человек и ученый. S+F-R=2

He was educated at home until entering public school at the age of 13. He was enrolled in the Polytechnique (a French College of Research) by age 16. He graduated with a degree in civil engineering by the time he was 21 years old. He returned home to begin pursuing mathematics. He held professorships at multiple colleges including the Faculte de Sciences and the College de France. He wrote a total of 789 papers explaining his mathematical discoveries in his lifetime.. His last words to his students were, «I will explain it all in my next memoir.» Молодший В.Н. О. Коши и революция в математическом анализе первой четверти XIX века //ИМИ, № 23. С. 32–55. F.Smithies, Cauchy and the Creation of Complex Function Theory, Cambridge University Press, 1997

Cauchy functional equation Cauchy formula for repeated integration Cauchy–Frobenius lemma Cauchy–Hadamard theorem Cauchy horizon Cauchy's integral formula Cauchy's integral theorem Cauchy interlacing theorem Cauchy–Kovalevskaya theorem Cauchy matrix Cauchy momentum equation Cauchy number Cauchy–Peano theorem Cauchy principal value Cauchy problem Cauchy product Cauchy's radical test Cauchy–Riemann equations Cauchy–Schwarz inequality Cauchy sequence Cauchy surface Cauchy's mean value theorem Cauchy stress tensor Cauchy's theorem (geometry) Cauchy's theorem (group theory) Euler-Cauchy stress principle Maclaurin–Cauchy test Binet–Cauchy identity Bolzano-Cauchy theorem Cauchy's argument principle Cauchy–Binet formula Cauchy boundary condition Cauchy condensation test Cauchy's convergence test Cauchy (crater) Cauchy determinant Cauchy distribution Cauchy's equation Cauchy–Euler equation  Differential equations  Mathematical physics  Algebra  Optics  Theory of elasticity  Number theory  Theory ofComplex functions  Mathematical Anaysis

 sum of the series  continuity and discontinuity points  derivative and differential  definition of function (formula is only one way of defining a function; a class of continuous functions (analytical as podklasss); functions of a complex variable  ordinary and improper definite integral Cours d'Analyse de l'École Royale Polytechnique of 1821 has a major impact in today's understanding of limits, continuity, and integrals In fact, some say that Cauchy gave the first reasonably successful rigorous foundation for calculus

Бернард Больцано Колядко В.И. Больцано.

 «Considerations of some aspects of elementary geometry» 1804  Contributions to a better grounded presentation of mathematics 1810  «The Binomial Theorem» 1816  «Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation;» 1817  «Three problems - rectification, calculating areas and volumes»  Theory of Science 1837 Mathematical works Bolzano also gave the first purely analytic proof of the fundamental theorem of algebra, which had originally been proven by Gauss from geometrical considerations

 The principles of axiomatic theory  methodological principles of evidence  ε-δ definition of a mathematical limit and continuity of the function  Supremum, infimum  Theory of real numbers  Necessary condition for convergence of the series with real terms  A sufficient condition for convergence of the series  basic principles of mathematical logic Рыхлик Карел Теория вещественных чисел в рукописном наследии Больцано // ИМИ, № 11. С. 515–532

Karl Theodor Wilhelm Weierstrass

From 1841 Weierstrass began his career as a qualified teacher of mathematics at the Pro- Gymnasium in Deutsch Krone (then – Braunsberg) Weierstrass published a full version of his theory of inversion of hyperelliptic integrals; he received his doctor’s degree, was approved by the professor of the Berlin Industrial Institute, and elected a member of the Berlin Academy of Sciences, became extraordinary professor at the University of Berlin. Main lecture author's courses (more than 70)  Introduction to the theory of analytic functions, including the theory of real numbers  The theory of elliptic functions, applications of elliptic functions to problems of geometry and mechanics  Theory of Abelian integrals and functions  Calculus of variations  Synthetic geometry Students of Karl Weierstrass: P.Bachmann, G.Cantor, F.G.Frobenius, A.Hurwitz, F.Klein, M.S.Lie, HG.Minkowski, G.Mittag-Leffler, H.Schwarz, O.Stolz; А.Н.Коркин, Н.В.Бугаев, В.Г.Имшенецкий, И.В.Слешинский, Б.Я.Букреев, Е.И.Золотарев, А.В.Бессель, В.П.Ермаков, Д.Ф.Селиванов, А.В.Васильев, М.А.Тихомандрицкий, П.М.Покровский

Humboldt University of Berlin, Sofia Vasilyevna Kovalevskaya «Weierstrass‘s lectures formed numerous students who have made the whole army, takes his direction, the army, he rushes forward, because he could not move himself everywhere» (A.Poincaré)

MATHEMATICAL ANALYSIS  theory of series based on the arithmetic  uniform convergence  analytic continuation of the function  doctrine of limit points  language "epsilon-delta" OTHER MATHEMATICAL ACHIEVEMENTS The calculus of variations: strong extremum conditions and sufficient conditions for an extremum, discontinuous solutions of the classical equations. Geometry: a theory of minimal surfaces, a contribution to the theory of geodesic lines. Linear algebra: the theory of elementary divisors. The theory of quadratic forms Projective geometry The evidence transcendence π, e

the Weierstrass function is an example of a real-valued function on the real line. The function has the property that it is continuous everywhere but differentiable nowhere

Bernhard Reimann

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale Holly Trochet, A History of Fractal Geometry, Перевод статьи Benoit Mandelbrot

Denition. Given any positive integer n, the aliquot part of n is the expression 1/n. J. Christopher Tweddle, Weierstrass's Construction of the Irrational Numbers Aggregates are the collections of these aliquot parts: {¼, ¼, ¼, ¼} (1), {1/7, 1/7} (2/7) Denition (Order). Let a and b be two aggregates. We define a ≤ b if, and only if, for all proper subaggregates a′ of transformations of a that contain finitely many aliquot parts, we can transform a′ to a′′ so that every aliquot part in a′′ occurs in b (which may likewise need to be transformed), including any multiplicities. Denition (Equality). Let a and b be two aggregates. Then a = b if, and only if, a ≤ b and b ≤ a.

Julius Wilhelm Richard Dedekind ( )

The idea behind a cut is that an irrational number divides the rational numbers into two classes (sets), with all the members of one class (upper) being strictly greater than all the members of the other (lower) class. For example, the square root of 2 puts all the negative numbers and the numbers whose squares are less than 2 into the lower class, and the positive numbers whose squares are greater than 2 into the upper class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in «Continuity and irrational numbers»

Georg Ferdinand Ludwig Philipp Cantor ( )

- rational numbers - fundamental sequences and their limits Даан-Дальмедико А., Пейффер Ж. Пути и лабиринты - М.: Мир, 1986 Бурбаки Н. Очерки по истории математики. - М.: ИИЛ, 1986 Definition. A real number is an equivalence class of Cauchy sequences of rational numbers. In 1872 Cantor defined irrational numbers in terms of convergent sequences of rational numbers; in 1873 he proved the rational numbers countable, i.e. they may be placed in one- one correspondence with the natural numbers. He also showed that the algebraic numbers, i.e. the numbers which are roots of polynomial equations with integer coefficients, were countable. Then Cantor had proved that the real numbers were not countable. A transcendental number is an irrational number that is not a root of any polynomial equation with integer coefficients. Liouville established in 1851 that transcendental numbers exist. Twenty years later, in this 1874 work, Cantor showed that in a certain sense 'almost all' numbers are transcendental by proving that the real numbers were not countable while he had proved that the algebraic numbers were countable. J J O'Connor and E F Robertson, The real numbers: Stevin to Hilbert

Giuseppe Peano (1858 – 1932) 1.There is a natural number 1. 2.Every natural number a has a successor, denoted by S(a) or. 3.There is no natural number whose successor is 1. 4.Distinct natural numbers have distinct successors: a = b if and only if S(a) = S(b). 5. If a property is possessed by 1 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers..