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Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.[1][2]

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.

Calculus of variations

deals with the representation of functions or signals as the superposition of basic waves.

Harmonic analysis

involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry.

Geometric analysis

the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.

Clifford 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.

p-adic analysis

which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.

Non-standard analysis

the study of which parts of analysis can be carried out in a computable manner.

Computable analysis

– analytical notions developed for stochastic processes.

Stochastic calculus

– applies ideas from analysis and topology to set-valued functions.

Set-valued analysis

the study of convex sets and functions.

Convex analysis

Idempotent analysis

Tropical analysis

which is built upon a foundation of constructive, rather than classical, logic and set theory.

Constructive analysis

which is developed from constructive logic like constructive analysis but also incorporates choice sequences.

Intuitionistic analysis

which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.

Paraconsistent analysis

which is developed in a smooth topos.

Smooth infinitesimal analysis

Analytic number theory

Analytic combinatorics

Continuous probability

in information theory

Differential entropy

Differential games

the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.

Differential geometry

Differentiable manifolds

Differential topology

Partial differential equations

Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau

Introductory Real Analysis, by , Sergei Fomin[28]

Andrey Kolmogorov

Differential and Integral Calculus (3 volumes), by [29][30][31]

Grigorii Fichtenholz

The Fundamentals of Mathematical Analysis (2 volumes), by [32][33]

Grigorii Fichtenholz

A Course Of Mathematical Analysis (2 volumes), by [34][35]

Sergey Nikolsky

Mathematical Analysis (2 volumes), by [36][37]

Vladimir Zorich

A Course of Higher Mathematics (5 volumes, 6 parts), by [38][39][40][41][42]

Vladimir Smirnov

Differential And Integral Calculus, by [43]

Nikolai Piskunov

A Course of Mathematical Analysis, by [44]

Aleksandr Khinchin

Mathematical Analysis: A Special Course, by [45]

Georgiy Shilov

Theory of Functions of a Real Variable (2 volumes), by [46][47]

Isidor Natanson

Problems in Mathematical Analysis, by [48]

Boris Demidovich

Problems and Theorems in Analysis (2 volumes), by , Gabor Szegö[49][50]

George Polya

Mathematical Analysis: A Modern Approach to Advanced Calculus, by [51]

Tom Apostol

Principles of Mathematical Analysis, by [52]

Walter Rudin

Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by [53]

Elias Stein

Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by [54]

Lars Ahlfors

Complex Analysis, by [55]

Elias Stein

Functional Analysis: Introduction to Further Topics in Analysis, by [56]

Elias Stein

Analysis (2 volumes), by [57][58]

Terence Tao

Analysis (3 volumes), by Herbert Amann, Joachim Escher[60][61]

[59]

Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov

[62]

Real and Functional Analysis, by [63]

Serge Lang

Constructive analysis

History of calculus

Hypercomplex analysis

Multiple rule-based problems

Multivariable calculus

Paraconsistent logic

Smooth infinitesimal analysis

Timeline of calculus and mathematical analysis

; Kolmogorov, A. N.; Lavrent'ev, M. A., eds. (March 1969). Mathematics: Its Content, Methods, and Meaning. Vol. 1–3. Translated by Gould, S. H. (2nd ed.). Cambridge, Massachusetts: The M.I.T. Press / American Mathematical Society.

Aleksandrov, A. D.

(1974). Mathematical Analysis (2nd ed.). Addison–Wesley. ISBN 978-0201002881.

Apostol, Tom M.

; Pfaffenberger, William Elmer (1981). Foundations of mathematical analysis. New York: M. Dekker.

Johnsonbaugh, Richard

(2002). "Mathematical analysis". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Springer-Verlag. ISBN 978-1402006098.

Nikol'skiĭ [Нико́льский], Sergey Mikhailovich [Серге́й Миха́йлович]

; Marcellini, Paolo; Sbordone, Carlo (1996). Analisi Matematica Due (in Italian). Liguori Editore. ISBN 978-8820726751.

Fusco, Nicola

Rombaldi, Jean-Étienne (2004). Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques (in French). . ISBN 978-2868836816.

EDP Sciences

(1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542358.

Rudin, Walter

(1987). Real and Complex Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542341.

Rudin, Walter

(PDF). Archived (PDF) from the original on 2007-04-19.

"Real Analysis – Course Notes"

Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis

by Jiri Lebl (Creative Commons BY-NC-SA)

Basic Analysis: Introduction to Real Analysis

Mathematical Analysis – Encyclopædia Britannica

Calculus and Analysis