Mathematical analysis

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This article is about a branch of mathematics. The words "mathematical analysis" are also used to describe the process or result of modeling and analyzing a phenomenon using mathematical techniques in general.

Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of 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 is equipped with a definition of "nearness" (a topological space) or more specifically "distance" (a metric space).

Contents 1 Motivation 2 History 3 Subdivisions 4 Notes 5 References

[edit] Motivation

The motivation for studying mathematical analysis in the wider context of topological or metric spaces is twofold:

• First, the same basic techniques have proved applicable to a wider class of problems (e.g., the study of function spaces).

• Second, and just as importantly, a greater understanding of analysis in more abstract spaces frequently proves to be directly applicable to classical problems. for example, in Fourier analysis functions are expressed in terms of certain infinite series (of complex exponentials or trigonometric functions). Physically, this decomposition amounts to reducing an arbitrary (sound) wave to its frequency components. The "weights" or coefficients of the terms in the Fourier expansion of a function can be thought of as components of a vector in an infinite dimensional space known as a Hilbert space. Study of functions defined in this more general setting thus provides a convenient method of deriving results about the way functions vary in space as well as time or, in more mathematical terms, partial differential equations, where this technique is known as separation of variables.

[edit] 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.^[3]

In India, the 12th century mathematician Bhaskara conceived of differential calculus, and gave examples of the derivative and differential coefficient, along with a statement of what is now known as Rolle's theorem. In the 14th century, mathematical analysis originated with Madhava in South India, who developed the fundamental ideas of the infinite series expansion of a function, the power series, the Taylor series, and the rational approximation of an infinite series. He developed the Taylor series of the trigonometric functions of sine, cosine, tangent and arctangent, and estimated the magnitude of the error terms created by truncating these series. He also developed infinite continued fractions, term by term integration, the Taylor series approximations of sine and cosine, and the power series of the radius, diameter, circumference, π, π/4 and angle θ. His followers at the Kerala School further expanded his works, up to the 16th century.

In Europe, during the latter half of the 17th century, Newton and Leibniz independently developed calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

In the 18th century, Euler introduced the notion of mathematical function.^[4] In the 19th century, Cauchy was the first to put calculus on a firm logical foundation by introducing the concept of the 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 "epsilon-delta" 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, in which a mathematician creates irrational numbers that serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers. 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.

[edit] 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^[5] 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.
• Differential geometry and topology, the application of calculus to abstract mathematical spaces that possess a complicated internal structure.
• 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.

[edit] Notes

1. ^ (Whittaker and Watson, 1927, Chapter III)
2. ^ Edwin Hewitt and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965
3. ^ (Smith, 1958)
4. ^ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, 17. 
5. ^ Carl L. Devito, "Functional Analysis", Academic Press, 1978

[edit] References

• Apostol, Tom M., Mathematical Analysis, 2nd ed. Addison-Wesley, 1974. ISBN 978-0201002881.
• Nikol'skii, S. M., "Mathematical analysis", in Encyclopaedia of Mathematics, Michiel Hazewinkel (editor), Springer-Verlag (2002). ISBN 1-4020-0609-8.
• Smith, David E., History of Mathematics, Dover Publications, 1958. ISBN 0-486-20430-8.
• Whittaker, E. T. and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927. ISBN 0521588073.

Major fields of mathematics v • d • e
Logic • Set theory • Algebra (Abstract algebra - Linear algebra) • Discrete mathematics • Combinatorics • Number theory • Analysis • Geometry • Topology • Applied mathematics • Probability • Statistics • Mathematical physics