# Mathematical analysis

Mathematical analysis. It begins to develop from the beginning of the rigorous formulation of infinitesimal calculus and studies concepts such as continuity, integration and differentiability of functions in different spaces and all supported by the concept of limit. This basic concept is a fundamental tool. both for continuity, derivative, definite integral, sequences and series sums.

## Summary

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• 1 Story
• 1 Taylor series
• 2 Mathematical calculation
• 3 Analytical geometry and mathematical analysis
• 4 Definition of the concept of function
• 5 Theory of integration
• 6 Theory of measurement
• 7 Functional analysis
• 2 Fields of mathematical analysis
• 3 References
• 4 Bibliography

## History

Greek mathematicians, such as the Cnidos and Archimedean Eudox , made informal use of the concepts of limit and convergence when they used the Exhaustion Method to calculate the area and volume of regions and solids. In fact, Number π | number π was approximated using the exhaustion method.

In India in the twelfth century mathematician Bhashara II conceived elements of differential calculus and the concept of what is now known as the theorem of Rolle .

In the fourteenth century , mathematical analysis originates with Madhava from Sangamagrama, in South Asia , who developed fundamental ideas such as the expansion of infinite (mathematical) series, power series, Taylor series, and the rational approximation of series infinite.

### Taylor series

In addition, he developed the Taylor series of trigonometric functions —sine, cosine, tangent—, and estimated the magnitude of the calculation errors by truncating these series. He also developed infinite continuous fractions, term-to-term integration, and the power series of pi . His disciples from the Kerala School continued their work until the 16th century .

### Mathematical calculation

The analysis in Europe originates from the 17th century , in which Newton and Godofredo Leibniz created differential and integral calculus. We now know that Newton developed infinitesimal calculus some ten years before Leibnitz. Each one independently, Leibniz’s work had a greater reception for the goodness and effectiveness of the concomitant symbology [1]

The latter did it in 1675 and published his work in 1684 , approximately twenty years before Newton decided to do the same with his works. Newton had communicated the novelty only to a few of his colleagues, and Halley’s prompting for Newton to publish his works earlier was of no avail .

This attitude served as the basis for creating an unpleasant controversy over the sponsorship of the idea; discussion that could have been avoided if another great mathematician, Fermat , had not also had the inexplicable habit of not making his works public.

### Analytical geometry and mathematical analysis

In a letter from Fermat to Roberval, dated October 22, 1636, both analytical geometry and mathematical analysis are clearly described. In that century and in the 18th century, certain topics on analysis such as calculus of variations, differential equations and partial derivative equations, Fourier harmonic analysis and generating functions were developed mainly for an application work. The calculation techniques were successfully applied in the approximation of discrete problems using the continuous ones.

### Definition of the function concept

Throughout the eighteenth century the definition of the concept of function was subject to debate among mathematicians. In the 19th century , Augustin Louis Cauchy was the first to establish the calculation on firm logical foundations by using the concept of succession of cauchy. He also initiated the formal theory of complex analysis. Simeon Poisson, Joseph Liouville, Jean-Baptiste Joseph Fourier, and others studied partial differential equations and harmonic analysis.

### Integration theory

At the middle of that century, Bernhard Riemann introduces his theory of integration. In the last third of the 19th century Karl Weierstrass led to the arithmeticization of analysis, since he thought that geometric reasoning was deceptive in nature, and introduced the definition ε-δ of limit.

Then mathematicians began to wonder if they were not assuming the existence of a certain continuum of real Numbers without proving their existence. Julius Wilhelm Richard Dedekind then constructs the real numbers using dedekind’s cuts. Around the same time, attempts to refine Riemann’s integration theorems led to the study of the “size” of discontinuity sets of real functions.

### Measurement theory

Also, “monster” functions (continuous functions nowhere, continuous functions but not differentiable at any point, space-filling curve, (Peano Curve) began to emerge. In this context Camille Jordan developed her measurement theory, Georg Cantor he did so with what is now called Set Theory, and René-Louis Baire proves the Baire Category Theorem.

### Functional analysis

In the early 20th century , calculus was formalized using set theory. Henri Léon Lebesgue solves the measurement problem, and David Hilbert introduces Hilbert spaces to solve integral equations. The idea of ​​normed vector spaces was in the making, and in the 1920s Stefan Banach created functional analysis.

## Fields of mathematical analysis

Mathematical analysis includes the following fields:

• Real analysis, that is, the formally rigorous study of derivatives and integrals of real-valued functions, which includes the study of limits, series and measurements.
• Functional analysis, which studies spaces and functions and introduces concepts such as Banach’s Space and Hilbert’s Space.
• Harmonic Analysis, which deals with the series by Jean-Baptiste Joseph Fourier and its abstractions.
• Complex analysis, which studies functions that go from the complex plane to itself and that are complex-differentiable, holomorphic functions.
• P-adic analysis, analysis in the context of p-adic numbers, which differs in an interesting and surprising way from its real and complex counterpart.
• Non-standard analysis, which investigates certain hyperreal numbers and their functions and gives a rigorous treatment of infinitesimal and infinitely large numbers.