Home | Site Map | Anoca.org

# Mathematical analysis

(mathematicalanalysis)

Analysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions . It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity , integration and differentiability in general settings.

## History

Historically, analysis originated in the 17th century , with theinvention of calculus by Newton and Leibniz . In the 17th and 18th centuries ,analysis topics such as the calculus of variations , differential and partial differential equations , Fourier analysis and generating functions were developed mostly in applied work. Calculus techniques were appliedsuccessfully to approximate discrete problems by continuous ones.

All through the 18th century the definition of the concept function was a subject of debate among mathematicians. In the 19th century , Cauchy was the first toput 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 ofanalysis by Weierstrass , who thought that geometric reasoning wasinherently 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 discontinuity sets ofreal functions.

Also, " monsters " ( nowhere continuous functions, continuous but nowhere differentiablefunctions, space-filling curves ) began to be created. In thiscontext, Jordan developed his theory of measure , Cantor developed what is now called naïve 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 space 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:

 Topics in mathematics related to structure Abstract algebra | Number theory | Algebraic geometry | Group theory | Monoids | Analysis | Topology | Linear algebra | Graphtheory | Universal algebra | Category theory | Order theory

 Topics in mathematics related to change Arithmetic | Calculus | Vector calculus | Analysis| Differential equations | Dynamical systems and chaos theory | List of functions

mathematcial analysis, functions, mathematical analsis, th, matheatical analysis, complex, mathematical aanlysis, study, matematical analysis, equations, mathemaitcal analysis, hilbert, mathemaical analysis, mathematics, mathematicla analysis, topics, mahtematical analysis, space, mathematica analysis, developed, mathematicala nalysis, fourier, mathematical anlaysis, cauchy, matehmatical analysis, mathematicians, mathematical analisis, set, mathematial analysis, created, mathematicalanalysis, baire, mathematical anaylsis, harmonic, mathmatical analysis, vector, mathmeatical analysis, plane, matheamtical analysis, concepts, mathematical analyiss, integration, , change, mathematica lanalysis, problem, athematical analysis, lebesgue, mtahematical analysis, integral, mahematical analysis, universal, mathematical aalysis, sbanach, mathematical analyis, normed, mathematical analyssi, axiomatic, mathematical anlysis, systems, mathematical analysi, jordan, mathematcal analysis, chaos, mathematical anaysis, cantor, mathematical analsyis, theorem, mathematicl analysis, nowadays, mathematical nalysis, ve, mathemtaical analysis, formalized, mathemtical analysis, standard, amthematical analysis, algebraic, mathematical naalysis, differentiable, mthematical analysis, filling, mathematiacl analysis, hyperreal, mathematical analyss, infinitely...

This article is completely or partly from Wikipedia - The Free Online Encyclopedia. Original Article. The text on this site is made available under the terms of the GNU Free Documentation Licence. We take no responsibility for the content, accuracy and use of this article.

Anoca.org Encyclopedia
0.01s