Mathematical analysis
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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.
History
Greek mathematicians such as Eudoxus and Archimedes made informal use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In the 12th century the Indian mathematician Bhaskara gave an example of what would now be called a "differential coefficient" and the basic idea behind what is now known as Rolle's theorem. The 14th century Indian mathematician Madhava of Sangamagramma expressed various trigonometric functions as infinite series, and estimated the magnitude of the error terms created by truncating these series.
In Europe, analysis originated in the 17th century, with the independent invention of calculus by Newton and Leibniz. In the 17th and 18th centuries, analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis and generating functions were developed mostly in applied work. Calculus techniques were applied successfully to approximate discrete problems by continuous ones.
All through the 18th century the definition of the concept of function was a subject of debate among mathematicians. In the 19th century, Cauchy was the first to put calculus on a firm logical foundation by introducing the concept of Cauchy sequence. He also started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis.
In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the ε-δ definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.
Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be created. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.
Subdivisions
Analysis is nowadays divided into the following subfields:
- 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.
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| Abstract algebra | Universal algebra | Graph theory | Category theory | Order theory | Model theory | Structural proof theory | |
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