Mathematical analysis

Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. These topics are often studied in the context of real numbers, complex numbers, and their functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or "distance" (a metric space). Mathematical analysis has its beginnings in the rigorous formulation of calculus.

Subdivisions

Analysis is nowadays divided into the following subfields:

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• Real analysis, the formally rigorous study of derivatives and integrals of real-valued functions. This includes the study of limits, power series and measures.
• Functional analysis 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 the complex plane which are complex differentiable.
• 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.

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 'straight' analysis is large.

Related Topics:
Differential equation - Dynamical systems

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