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'''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 the invention 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 applied successfully 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 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
discontinuity sets 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
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:
*
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.
'''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.
Non-standard analysis 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.
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'''Topics in
mathematics related to structure'''
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|align=center|
Abstract algebra |
Number theory |
Algebraic geometry |
Group theory |
Monoids |
Analysis |
Topology |
Linear algebra |
Graph theory |
Universal algebra |
Category theory |
Order theory
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{| style="margin:0 auto;" align=center width=80% id=toc
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'''Topics in
mathematics related to change'''
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|align=center|
Arithmetic |
Calculus |
Vector calculus |
Analysis |
Differential equations |
Dynamical systems and chaos theory |
List of functions
|}
de:Analysis
es:Anlisis matemtico
fi:Analyysi
fr:Analyse (mathmatiques)
ja:解析学
nl:Analyse (wiskunde)
pl:Analiza matematyczna
sr:Анализа
