The name of analysis was first introduced into mathematics by Newton, because calculus is regarded as an extension of algebra, and "analysis" is synonymous with "algebra". Today, although it refers to a wider range, it is still only a summary of the similarities and differences between the methods of the disciplines involved, and it is increasingly difficult to completely distinguish it from the methods of geometry and algebra.
The oldest and most basic part of analysis is mathematical analysis. It was founded by Newton and Leibniz in the17th century in order to solve the problems raised by production and science at that time through the efforts of many mathematicians. However, the work of establishing a strict logical basis for analysis was not completed until the19th century. Since then, mathematical analysis has become a complete mathematical discipline. Mathematical analysis is the earliest discipline to systematically study functions. Although it is basically a kind of function with quite good properties-continuous function on interval, it is of great significance both in theory and in application. Theoretically, mathematical analysis is the same foundation of analytical disciplines, and it is also their birthplace. Among many branches of modern analysis, some were once part of mathematical analysis (such as variational method, Fourier analysis and even complex variable function theory). ), while others have deeply studied and extended some problems in mathematical analysis, such as real variable function theory, functional analysis, manifold analysis and so on, after establishing a complete mathematical analysis system for various needs.
From the end of 19 to the beginning of the 20th century, due to the development of some branches of mathematics (such as Fourier analysis) and physics, the concept of function integrable in mathematical analysis gradually became clear, and it was further required to extend the integral to a wider range of functions, hoping that the integral operation would be more flexible and convenient. At the same time, in the continuous discussion of the relationship between basic concepts in mathematical analysis (such as whether the inverse operation of differential and integral in general sense is established), people also feel it necessary to break through the limitations of mathematical analysis.
In this respect, Lebesgue's integral theory put forward in the early 20th century is of great significance, and the central content of the theory of real variable function is Lebesgue's integral theory. As a generalization of Riemann integral, Lebesgue integral not only has a wide range of integrable functions, but also has good properties such as countable additivity. The condition of finding the limit under the integral sign is also relaxed, and its theory has been fully developed, so it is more suitable for the needs of various branches of mathematics and physics. Because of the completeness of the space (function class) of Lebesgue integrable function, it occupies an important position in mathematical theory that Riemann integral can not have. Like mathematical analysis, real variable function theory also studies the basic properties of functions, such as continuity, differentiability and integrability. However, due to the application of set theory, it is possible to study functions on general point sets, so the research results are broader and more perfect than mathematical analysis. Therefore, the theory of real variable function has also become one of the * * * same foundations of various branches of analysis (especially modern branches such as functional analysis). On the question of whether differential and integral are reciprocal operations, the result of Lebesgue integral is one step further than that of Riemann integral. However, in order to solve this problem completely, later, some people put forward a variety of broader integration theories, such as Danjoy integral and Peron integral. Finally, the generalized Dangruwa integral (19 16) gives a positive answer to the above questions. However, these integrals are of great significance in specific theoretical problems, which are far less universally applicable than Lebesgue integrals. Lebesgue integral is based on Lebesgue measure, which further develops into abstraction, promotes the systematic study of measure and forms an independent discipline, which is measure theory. Measurement is a generalization of the concepts of area and volume, which are always closely related to the concept of integral. The thought and theory of measure theory are very important and useful in modern analysis.