   #copyright

Mathematical analysis

2007 Schools Wikipedia Selection. Related subjects: Mathematics

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

   Analysis is a branch of mathematics that depends upon the concepts of
   limits and convergence. It 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 more
   specifically "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 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 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 concept of function and it
   became 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 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. 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

   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 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.
     * 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 'straight' analysis is
   large.
   Retrieved from " http://en.wikipedia.org/wiki/Mathematical_analysis"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
   of authors and sources) and is available under the GNU Free
   Documentation License. See also our Disclaimer.
