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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.[(Whittaker and Watson, 1927, Chapter III)] It also includes the theories of Derivative|differentiation, Integral|integration and Measure (mathematics)|measure, Series (mathematics)|infinite series,[Edwin Hewitt and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965] and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex function (mathematics)|functions. However, they can also be defined and studied in any space (mathematics)|space of mathematical objects that has a definition of ''nearness'' (a topological space) or, more specifically, ''distance'' (a metric space).
History
Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno of Elea|Zeno's Zeno's paradoxes|paradox of the dichotomy.[(Smith, 1958)] In Indian mathematics|India, the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem.
In the 14th century, Madhava of Sangamagrama developed series (mathematics)|infinite series expansions, like the power series and the Taylor series, of functions such as Trigonometric functions|sine, Trigonometric functions|cosine, trigonometric functions|tangent and Inverse trigonometric functions|arctangent. Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century.
In Europe, during the later half of the 17th century, Isaac Newton|Newton and Gottfried Leibniz|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 differential equation|ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete mathematics|discrete problems by continuous ones.
In the 18th century, Leonhard Euler|Euler introduced the notion of function (mathematics)|mathematical function.[*] In the 19th century, Augustin Louis Cauchy|Cauchy helped 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. Siméon Denis Poisson|Poisson, Joseph Liouville|Liouville, Joseph Fourier|Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Karl Weierstrass|Weierstrass, developed the modern notion of mathematical rigor, thus founding the field of mathematical analysis (at least in the modern sense).
In the middle of the century Bernhard Riemann|Riemann introduced his theory of integral|integration. The last third of the 19th century saw the arithmetization of analysis by Karl Weierstrass|Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the (ε, δ)-definition of limit|"epsilon-delta" definition of limit of a function|limit.
Then, mathematicians started worrying that they were assuming the existence of a Continuum (set theory)|continuum of real numbers without proof. Richard Dedekind|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 metric space|complete set: the continuum of real numbers. Around that time, the attempts to refine the theorems of Riemann integral|Riemann integration led to the study of the "size" of the set of Classification of discontinuities|discontinuities of real functions.
Also, "pathological (mathematics)|monsters" (nowhere continuous functions, continuous but Weierstrass function|nowhere differentiable functions, space-filling curves) began to be created. In this context, Camille Jordan|Jordan developed his theory of Jordan measure|measure, Georg Cantor|Cantor developed what is now called naive set theory, and René-Louis Baire|Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Henri Lebesgue|Lebesgue solved the problem of measure, and David Hilbert|Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Stefan Banach|Banach created functional analysis.
Subdivisions
Mathematical analysis includes the following subfields.
- Differential equations
- Real analysis, the rigour#Mathematical rigour|rigorous study of derivatives and integrals of functions of real variables. This includes the study of sequences and their limit of a sequence|limits, series (mathematics)|series.
- Real analysis on time scales - a unification of real analysis with calculus of finite differences
- measure (mathematics)|Measure theory - given a set, the study of how to assign to each suitable subset a number, intuitively interpreted as the size of the subset.
- Functional analysis[Carl L. Devito, "Functional Analysis", Academic Press, 1978] 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|''p''-adic analysis, the study of analysis within the context of p-adic number|''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 rigour#Mathematical rigour|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.
- Set-valued analysis - applies ideas from analysis and topology to set-valued functions.
- Stochastic calculus - analytical notions developed for stochastic processes
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.
Applications
- Analytic number theory
- Analytic combinatorics
- Continuous probability
- Differential entropy in information theory
- Differential games
Topological spaces, metric spaces
The motivation for studying mathematical analysis in the wider context of topology|topological or metric spaces is three-fold:
- Firstly, the same basic techniques have proved applicable to a wider class of problems (e.g., the study of functional analysis|function spaces).
- Secondly, 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 a certain infinite sum of trigonometric functions. Thus Fourier analysis might be used to decompose a sound into a unique combination of pure tones of various pitches. The "weights", or coefficients, of the terms in the Fourier expansion of a function can be thought of as components of a vector (mathematics)|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.
- Thirdly, the conditions needed to prove the particular result are stated more explicitly. The analyst then becomes more aware exactly what aspect of the assumption is needed to prove the theorem.
See also
- Method of exhaustion
- Non-classical analysis
- Smooth infinitesimal analysis
Notes
References
- Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill Publishing Co.; 3Rev Ed edition (September 1, 1976), ISBN 978-0070856134.
- 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.
*
- E. T. Whittaker|Whittaker, E. T. and G. N. Watson|Watson, G. N., ''Whittaker and Watson|A Course of Modern Analysis'', fourth edition, Cambridge University Press, 1927. ISBN 0521588073.
- Jean-Étienne Rombaldi, ''Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques''
- K.G. Binmore, ''The foundations of analysis: a straightforward introduction''
- Richard Johnsonbaugh & W. E. Pfaffenberger, ''Foundations of mathematical analysis''
- Aleksandrov, A. D., Kolmogorov, A. N., Lavrent'ev, M. A. (Ed.), Translated by S. H. Gould, K. A. Hirsch and T. Bartha. Translation edited by S. H. Gould; "Mathematics, its Content, Methods, and Meaning", The M.I.T Press; Published in cooperation with the American Mathematical Society, Second Edition, Fourth Printing, 1984 Cambridge, Massachusetts, Library of Congress Card Number: 64-7547.
Web pages
- Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
- Basic Analysis: Introduction to Real Analysis by Jiri Lebl
Category:Mathematical analysis|*
simple:Mathematical analysis
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