Principles of Mathematical Analysis

Euler created the concept of mathematical function in the 18th century. When Bernard Bolzano proposed the contemporary definition of continuity in 1816, real analysis began to emerge as a separate field, but his work was not well known until the 1870s. By rejecting the notion of the generality of algebra, which had been widely used in earlier work, particularly by Euler, Cauchy began to set calculus on a sound logical foundation in 1821. Cauchy instead defined calculus in terms of geometric concepts and infinitesimals. As a result, an infinitesimal change in x had to correspond to an infinitesimal change in y in his notion of continuity. He also established the Cauchy sequence notion and the formal theory of complex analysis. Partial differential equations and harmonic analysis were addressed by Poisson, Liouville, Fourier, and others. These and other mathematicians’ contributions, such as Weierstrass’, helped to construct the (,)-definition of the limit approach, which helped to establish the present area of mathematical analysis.

Riemann introduced his theory of integration in the mid-nineteenth century. Weierstrass, who believed that geometric reasoning was intrinsically misleading and introduced the “epsilon-delta” notion of limit in the last part of the century, arithmetic analysis. Mathematicians became concerned that they were presuming the existence of a continuous series of real numbers without proof. Dedekind then built the real numbers using Dedekind cuts, in which irrational numbers are explicitly defined and used to fill “gaps” between rational numbers, resulting in a full set: the continuum of real numbers, which Simon Stevin had already produced in terms of decimal expansions. Around that time, attempts to improve Riemann integration theorems led to the study of the “size” of the set of real-function discontinuities.

Main branches of analysis

Real analysis

Real analysis (also known as the theory of real-valued functions of a real variable) is an area of mathematics that deals with real numbers and real-valued functions of a real variable. It focuses on the analytic properties of real functions and sequences, such as real-number sequence convergence and limits, real-number calculus, continuity, smoothness, and related features of real-valued functions.

Complex analysis

Complex analysis is a branch of mathematical analysis that studies the functions of complex numbers. It is also known as the theory of functions of a complex variable. It has applications in algebraic geometry, number theory, and applied mathematics, as well as in physics, such as hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and, most notably, quantum field theory.

The analytic functions of complex variables are of particular interest in complex analysis (or, more generally, meromorphic functions). Complex analysis is commonly used in physics to solve two-dimensional issues since the separate real and imaginary components of any analytic function must meet Laplace’s equation.

Functional analysis

The study of vector spaces equipped with some kind of limit-related character (e.g. inner product, norm, topology, etc.) and the linear operators acting on these spaces and preserving these structures in an appropriate manner constitutes the core of the functional analysis. The study of function spaces and the specification of attributes of function transformations like the Fourier transform as transformations defining continuous, unitary, and other operators between function spaces are the historical foundations of functional analysis. This viewpoint proved to be especially beneficial for studying differential and integral equations.

Infinite series

In the manipulation of infinite series, such as 

             1/2 + 1/4 + 1/8 …….(1) 

going forever, similar paradoxes exist. This specific series is rather innocuous, with a value of exactly 1. Consider the partial sums created by halting after a finite number of terms to discover why this is the case. The partial sum approaches 1 as the number of terms increases. By including enough terms, it can be made as near to 1 as desired. Furthermore, the facts above are true exclusively for integer 1. As a result, it makes sense to define the infinite sum as 1.

Other infinite series, such as 

1-1 + 1-1 + 1-1 +…….  (2) 

If the terms are arranged in one direction, (1-1) + (1-1) + (1-1) +, the total looks to be 0 + 0 + 0 + = 0.

However, if the elements are arranged differently, 

1 + (-1 + 1) + (-1 + 1) + (-1 + 1) +………  

the sum seems to be 1 + 0 + 0 + 0 + 0 + 1 = 1.

It would be a mistake to assume that 0 Equals 1. The conclusion is that infinite series do not always follow typical algebraic principles, such as those that allow for arbitrary word regrouping.

The partial sums show the difference between series (1) and (2). The partial sums of (1) approach a single fixed value, 1, more and more. (2)’s partial sums fluctuate between 0 and 1, causing the series to never settle. All other series is said to diverge. A series that does settle down to some definite value as more and more terms are added is said to converge, and the value to which it converges is known as the limit of the partial sums.

Conclusion

Limits and related notions, such as differentiation, integration, measure, sequences, series, and analytic functions, are dealt with in the discipline of mathematics known as analysis. Riemann introduced his theory of integration in the mid-nineteenth century. Partial differential equations and harmonic analysis were addressed by Poisson, Liouville, Fourier, and others. Real analysis (also known as the theory of real-valued functions of a real variable) is an area of mathematics that deals with real numbers and real-valued functions of a real variable.

Complex analysis is a branch of mathematical analysis that studies the functions of complex numbers. It is also known as the theory of functions of a complex variable. 

The study of function spaces and the specification of attributes of function transformations like the Fourier transform as transformations defining continuous, unitary, and other operators between function spaces are the historical foundations of functional analysis.