The establishment of mathematical analysis began with the pioneering work represented by Newton (I.) and Leibniz (G.W.) in the17th century, and was completed in the19th century. Calculus represented by Cauchy (A.-L.) and Wilstrass (T.W.) and its related contents have been called analysis since Newton. Since then, the field of calculus has been expanding, but many mathematicians still use this name. Today, although many contents have been separated from calculus and become independent disciplines, people still call it analysis. Mathematical analysis is also called analysis for short.
newton
The research object of mathematical analysis is function, and the basic behavior of function is studied from both local and whole aspects, thus forming the basic content of differential calculus and integral calculus. Differential calculus studies the local characteristics of functions, such as the rate of change. Derivative and differential are its main concepts, and the process of finding derivative is differential method. Around the nature, calculation and direct application of derivative and differential, the main content of differential calculus is formed. Integral studies the total effect of small changes (especially non-uniform changes) as a whole. Its basic concepts are original function (anti-derivative) and definite integral, and the process of finding integral is integral method. The nature, calculation, popularization and direct application of integral constitute the whole content of integral. Newton and Leibniz's outstanding contribution to mathematics lies in that around 1670, they summarized a series of basic laws of finding derivatives and integrals, and found that finding derivatives and integrals are two reciprocal operations, and reflected this reciprocal relationship through the famous formula named after them later, thus combining differential calculus and integral calculus, which were originally independently developed, into a new discipline-calculus. Thanks to the contributions of them and some later scholars (especially Euler (L.)), the differential integral method, which was originally known by only a few mathematicians and could only deal with some specific problems quite difficultly, has become a mechanical method that ordinary people can master with a little training, opening the door to its wide application in the field of science and technology, and its influence is immeasurable. Therefore, the appearance and development of calculus is regarded as one of the epoch-making events in the history of human civilization. Compared with integral, infinite series is also the superposition and accumulation of tiny quantities, but it takes discrete form (integral is continuous form). Therefore, in mathematical analysis, infinite series and calculus have always been inseparable and complementary. In history, infinite series has been used for a long time, but only after it became a part of mathematical analysis did it get real development and wide application.
Euler
The basic method of mathematical analysis is limit method or infinitesimal analysis. The word infinitesimal analysis appeared in the title of the world's first calculus textbook published by L'Hospital (G.-F.-A. de) in Paris in 1696 and Euler's two-volume book "Communication Calculus and Elementary Analysis" published in 1748. In the early stage of the development of calculus, this new method showed great power and obtained a lot of important results. Many new branches of mathematics related to calculus, such as variational methods, differential equations, differential geometry and the theory of complex variable functions, were developed in the early18-19th century. However, the initial analysis was relatively rough, and mathematicians inspired by the power of the new method often ignored the logical basis of deduction and used intuitive inference and contradictory reasoning, so that people generally doubted the rationality of this method throughout the18th century. These doubts are largely caused by the meaning and usage of infinitesimal that was often used at that time. Random use and explanation of infinitesimal leads to confusion and mystery. Many people participated in the debate about the essence of infinitesimal, and some of them, such as Lagrange (J.-L.), tried to exclude infinitesimal sum limit and algebraic calculus. Debate gradually clarified the concepts of function and limit. More and more mathematicians realize that the concept of mathematical analysis must be distinguished from its prototype in the objective world and human intuition.
Cauchy
Therefore, from the19th century, a critical transition period of new mathematical analysis characterized by arithmeticization of analysis (making analysis a deductive system like arithmetic) began. Cauchy's Analysis Course published by 182 1 is a sign of rigorous analysis. In this book, Cauchy established a limit close to the modern form, and defined infinitesimal as a variable tending to zero, thus ending the century-old debate. On the basis of limit, Cauchy defined the continuity, derivative, integral and convergence of series of continuous functions (later, Porzano (. Furthermore, Dirichlet (P.G.L.) 1837 put forward the strict definition of function, and Weierstrass introduced the ε-δ definition of limit. Basically, the arithmetic of analysis is realized, and the analysis is liberated from the limitation of geometric intuition, thus dispelling the mysterious cloud hanging over calculus in the17-18th century.
Then, on this basis, Riemann ((G.F.) B.) and Dabu ((J.) G.) established the strict integration theory of bounded functions in 1854 and 1875, and Dai Dejin (Dede) was in the second half of 19 century.