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
– 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
Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by [54]
Lars Ahlfors
Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov
[62]
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.
(1981) [1981]. The foundations of analysis: a straightforward introduction. Cambridge University Press.
Binmore, Kenneth George
; 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
; Watson, George Neville (1927-01-02). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th ed.). Cambridge: at the University Press. ISBN 0521067944. (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992)