Analysis

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.

These theories are typically investigated in relation to real and complex numbers and functions. Calculus, which includes the fundamental concepts and tools of analysis, evolved into analysis. Although analysis is distinct from geometry, it can be applied to any space of mathematical objects with a notion of closeness (a topological space) or specified distances between items (a metric space).

Important concepts

A metric space is a set in mathematics where a notion of distance (called a metric) between set components is defined.

The real line, the complex plane, Euclidean space, various vector spaces, and integers are the most often used metric spaces for analysis. Measure theory (which describes size rather than distance) and functional analysis are two examples of non-metric research (which studies topological vector spaces that need not have any sense of distance).

Sequence and limits

An ordered list is referred to as a sequence. It has members, just like a set (also called elements, or terms). Unlike a set, order matters, and the same items might appear several times in the sequence at various points. A sequence can be described more explicitly as a function whose domain is a countable completely ordered set, such as the natural numbers.

Convergence is one of the most significant features of a series. If a series has a limit, it is said to converge. Informally, if n becomes very large, a (singly-infinite) sequence has a limit if it approaches some point x, termed the limit. That is, the distance between a and x approaches 0 for an abstract sequence (a_n) (with n going from 1 to infinity).

        lim _n→∞ a_n = x

Main branches

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, and continuity, smoothness, and related features of real-valued functions.

Complex analysis

Complex analysis is a branch of mathematics that studies 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 physics, such as hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and, most notably, quantum field theory.

Complex analysis is concerned with the analytic functions of complex variables in particular (or, more generally, meromorphic functions). Complex analysis is commonly used for two-dimensional physics problems because 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 boundary structure (e.g. inner product, norm, topology, etc.) and the linear operators acting on these spaces and preserving these structures in an appropriate manner is at the heart of functional analysis. The study of function spaces and the specification of attributes of function transformations such as the Fourier transform as transformations defining continuous, unitary, and other operators between function spaces are the historical foundations of functional analysis. This perspective was particularly handy while studying differential and integral equations.

Harmonic analysis

Harmonic analysis is an area of mathematics that studies and generalises the concepts of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). It has evolved into a huge field with applications in number theory, representation theory, signal processing, quantum mechanics, tidal analysis, and neuroscience over the last two centuries.

Applications

Analysis techniques can also be found in other fields, such as:

Physical science

Applied analysis, particularly differential equations, underpins the great majority of classical mechanics, relativity, and quantum mechanics. Newton’s second law, the Schrödinger equation, and the Einstein field equations are examples of major differential equations.

In quantum mechanics, functional analysis is also important.

Signal processing

Fourier analysis can isolate specific components of a compound waveform, concentrating them for easy detection or elimination, when processing signals like as audio, radio waves, light waves, seismic waves, and even photographs. Fourier-transforming a signal, manipulating the Fourier-transformed data in a straightforward fashion, and reversing the transformation are all part of a vast family of signal processing techniques.

Conclusion

Analysis is a discipline of mathematics that studies limits and associated notions like differentiation and integration. These theories are typically investigated in relation to real and complex numbers and functions. Although analysis is distinct from geometry, it can be applied to any space of mathematical objects with a notion of closeness (a topological space) or specified distances between items (a metric space).

The real line, the complex plane, Euclidean space, various vector spaces, and integers are the most often used metric spaces for analysis.

A sequence can be described more explicitly as a function whose domain is countable completely. 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.

The study of vector spaces equipped with some kind of boundary structure (e.g. inner product, norm, topology, etc.) and the linear operators acting on these spaces and preserving these structures in an appropriate manner is at the heart of functional analysis.