THE DEVELOPMENT OF MATHEMATICAL ANALYSIS AND ITS IMPORTANCE IN THE STUDY OF SCIENCES

ABSTRACT

This article examines the development of mathematical analysis as a fundamental discipline that plays a key role in the study and progress of scientific knowledge. The article analyzes the history of the emergence and evolution of mathematical analysis, starting with the works of great scientists of antiquity and ending with modern achievements. Special attention is paid to the influence of mathematical analysis on the development of sciences such as physics, engineering, economics and biology. It is shown how the principles and methods of mathematical analysis contribute to a deep understanding of natural phenomena, economic patterns and technological processes. The article highlights the interdisciplinary importance of mathematical analysis and its undeniable contribution to the comprehensive development of the scientific world.

Keywords: mathematical analysis, history of mathematics, physics, economics, biology, technological processes, interdisciplinary significance.

Mathematics combines elements of science and art, while its foundation is theoretical mathematics. Theoretical mathematics studies the limits of mathematical concepts and pure intelligence. It is often described as a segment of mathematical research carried out without direct attention to immediate practical applicability. However, what seems abstract today may find its application in the future. Examples of such applications include the financial sector and cryptography, where theoretical mathematics finds significant use.

In general, analytical mathematics deals with approximating certain mathematical structures, such as numbers or functions, with the help of other, more understandable or user-friendly structures. For example, to determine the first few digits of pi, you will most likely represent pi as the limit of a sequence of numbers known to you. And if we consider another example, the sequence of n factorials! It has visual appeal, but approximations are often used to analyze its growth. So, the growth of n! it can be visually represented using the famous Stirling formula, which gives an approximation convenient for calculations:

If you've already dealt with the basics of calculus, you've probably realized one of the central principles of analysis: the possibility of approximating differentiable functions by linear ones at the local level. This principle can be considered a fundamental starting point for analysis, going back three hundred years ago. Nevertheless, modern analysis differs significantly from the one that was studied even a hundred years ago.

Analysis, like other critical areas of mathematics, has continued to evolve and today finds applications in science, engineering, and economics, where, for example, the financial sector provides a significant number of jobs for mathematicians through analytical methods.

The relevance of the analysis today is such that it becomes almost impossible to give an exhaustive review due to the huge number of accumulated results. The key areas of interest to specialists in the field of theoretical mathematics include: real analysis, Fourier analysis (including wavelets), functional analysis, theory of operators and algebras, harmonic analysis, probability theory and measure. Differential equations are also an important part of analysis, actively studied in applied mathematics, including at the University of Waterloo. Ancient Greek scientists such as Eudoxus and Archimedes had already used the concepts of limits and convergence in their methods of exhaustion to determine areas and volumes.

In India, in the 12th century, mathematician Bhaskara developed the foundations of differential calculus, presenting examples of derivatives and differential analysis, and formulated the principle now known as Rolle's theorem.

By the 14th century in South India, Madhava had laid the foundations for mathematical analysis, developing ideas for decomposing functions into infinite series, including power series and Taylor series, as well as rational approximations. He derived Taylor series for trigonometric functions and estimated the errors from their truncation. Madhava also advanced methods of infinite continued fractions and term integration, as well as the approximation of sine and cosine.

In Europe, by the end of the 17th century, Newton and Leibniz had formed the foundations of calculus, which, expanded by applied research in the 18th century, evolved into such areas of analysis as calculus of variations, which, under the influence of applied work that continued in the 18th century, turned into topics of analysis such as calculus of variations, ordinary equations and partial differential equations., Fourier analysis and generating functions. During this period, calculus methods were used to approximate discrete problems by continuous ones.

In the 18th century, Euler proposed the concept of function, which sparked debate among scientists. In the 19th century, Cauchy laid the foundations of calculus on a strict logical basis, introducing the concept of the Cauchy sequence and beginning the development of a formal theory of complex analysis. Scientists like Poisson, Liouville, and Fourier have been actively working on partial derivatives and harmonic analysis. At the same time, Riemann proposed his own integration method.

By the end of the 19th century, Weierstrass had made a significant contribution by arithmetizing the analysis and proposing the definition of the limit through the concepts of «epsilon» and «delta». This sparked discussions about the existence of a continuum of real numbers, to which Dedekind responded by suggesting the construction of real numbers through his cuts. These reflections contributed to the refinement of the theories of Riemann integration and the study of the «size» of sets of discontinuities of real functions.

During this period, the so-called «mathematical monsters» arose, for example, functions that are continuous everywhere, but nowhere differentiable, and curves that fill space. Jordan developed measure theory, Cantor developed naive set theory, and Baer proved the category theorem. At the beginning of the 20th century, calculus was formalized through the axiomatic theory of sets. Lebesgue solved the questions of measure, and Hilbert introduced the concept of Hilbert spaces for solving integral equations. In the 1920s, Banach laid the foundations of functional analysis.

The subdomains of mathematical analysis include:

• Real analysis, which studies derivatives and integrals of functions of real variables, including the theory of sequences, series, and measures.
• Functional analysis, focused on the study of functional spaces, including Banach and Hilbert spaces.
• Harmonic analysis, working with Fourier series and their generalizations.
• Complex analysis, analyzing the functions of the complex plane.
• P-adic analysis, which studies p-adic numbers.
• Non-standard analysis exploring hyperreal numbers and the concepts of infinitesimals and large numbers.
• Numerical analysis, which develops algorithms for approximating continuous mathematics problems.

Classical analysis traditionally includes work that does not use functional analysis methods, and is often associated with more traditional areas in mathematics.

The study of differential equations interacts with other fields, for example, with the theory of dynamical systems, although it remains closely related to direct analysis.

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