# Mathematical analysis

Branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

- Mathematical analysis57 related topics

## Rigour

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

During the 19th century, the term "rigorous" began to be used to describe increasing levels of abstraction when dealing with calculus which eventually became known as mathematical analysis.

## Bernard Bolzano

Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views.

Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views.

To this end, he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik (1810), Der binomische Lehrsatz (1816) and Rein analytischer Beweis (1817).

## Representation theory

Branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

Branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups.

## Limit (mathematics)

Value that a function approaches as the input (or index) approaches some value.

Value that a function approaches as the input (or index) approaches some value.

Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

## Generality of algebra

Phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange, particularly in manipulating infinite series.

Phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange, particularly in manipulating infinite series.

In works such as Cours d'Analyse, Cauchy rejected the use of "generality of algebra" methods and sought a more rigorous foundation for mathematical analysis.

## Applied mathematics

Application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry.

Application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry.

Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability.

## George Pólya

Hungarian mathematician.

Hungarian mathematician.

He remained a Professor Emeritus at Stanford for the rest of his career, working on a range of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability.

## Nonstandard analysis

Fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

Fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

More generally, nonstandard analysis is any form of mathematics that relies on nonstandard models and the transfer principle.

## René Descartes

French philosopher, mathematician, scientist and lay Catholic who invented analytic geometry, linking the previously separate fields of geometry and algebra.

French philosopher, mathematician, scientist and lay Catholic who invented analytic geometry, linking the previously separate fields of geometry and algebra.

He is credited as the father of analytic geometry, the bridge between algebra and geometry—used in the discovery of infinitesimal calculus and analysis.

## Zeno's paradoxes

Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c.

Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c.

Today's analysis achieves the same result, using limits (see convergent series).