Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.^[1] It also includes the theories of differentiation, integration and measure, infinite series,^[2] and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. However, they can also be defined and studied in any space of mathematical objects that has a definition of nearness (a topological space) or, more specifically, distance (a metric space).

Subdivisions

Mathematical analysis includes the following subfields.

• Real analysis, the rigorous study of derivatives and integrals of functions of real variables. This includes the study of sequences and their limits, series, and measures.
• Functional analysis^[7] studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.
• Harmonic analysis deals with Fourier series and their abstractions.
• Complex analysis, the study of functions from the complex plane to itself which are complex differentiable (that is, holomorphic).
• Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
• p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
• Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers. It is normally classed as model theory.
• Numerical analysis, the study of algorithms for approximating the problems of continuous mathematics.

Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called hard analysis; it also naturally refers to the more traditional topics. The study of differential equations is now shared with other fields such as dynamical systems, though the overlap with conventional analysis is large.

Vahid Damanafshan

Reference: Wikipedia