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| Analysis (mathematics) | |
|---|---|
| Name | Analysis |
| Field | Mathematics |
| Introduced | 17th century |
| Notable people | Isaac Newton, Gottfried Wilhelm Leibniz, Augustin-Louis Cauchy, Karl Weierstrass, Bernhard Riemann, Henri Lebesgue, Georg Cantor, Alan Turing, René Descartes, Joseph Fourier, Leonhard Euler, Niels Henrik Abel, Évariste Galois, David Hilbert, Emmy Noether, John von Neumann, Felix Klein, Henri Poincaré, Sofia Kovalevskaya |
Analysis (mathematics) is a branch of mathematics concerned with limits, continuity, differentiation, integration, measure, infinite series, and the rigorous study of change and approximation. It grew from the foundational work of Isaac Newton and Gottfried Wilhelm Leibniz in calculus and was formalized through contributions by Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann. Analysis underpins modern work in David Hilbert's functional analysis, Henri Lebesgue's measure theory, and applications across Alan Turing's computation theory and John von Neumann's operator algebras.
The historical development traces from the methods of Isaac Newton and Gottfried Wilhelm Leibniz through the 18th-century work of Leonhard Euler and Joseph Fourier, to 19th-century rigorization by Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann. 19th-century advances involved Georg Cantor's set theory, Niels Henrik Abel's and Évariste Galois's algebraic insights, and Henri Poincaré's qualitative methods, while 20th-century formalization engaged David Hilbert, Emmy Noether, and John von Neumann. Schools in Paris, Berlin, and Cambridge shaped pedagogy and research culture alongside institutions like the École Normale Supérieure and the Kaiser Wilhelm Society.
Foundational work established rigorous definitions of limit, continuity, and convergence through axiomatization efforts by Karl Weierstrass and logical formalism influenced by Gottlob Frege and Bertrand Russell. Measure theory and integration were reconstructed by Henri Lebesgue and later extended in the work of Alfréd Haar and Andrey Kolmogorov for probability, while topology and metric spaces formalized by Felix Hausdorff and Maurice Fréchet provide the setting for modern analysis. Functional analysis codified by David Hilbert and Stefan Banach defines spaces of functions and operators central to the discipline.
Core topics include differential calculus as developed by Isaac Newton and Gottfried Wilhelm Leibniz, integral calculus refined by Bernhard Riemann and Henri Lebesgue, and series and sequence convergence studied by Augustin-Louis Cauchy and Karl Weierstrass. Real analysis and complex analysis build on concepts advanced by Bernhard Riemann and Augustin-Louis Cauchy, while measure theory and probability rest on Andrey Kolmogorov and Henri Lebesgue. Functional analysis and operator theory, influenced by John von Neumann and Stefan Banach, treat infinite-dimensional problems, and harmonic analysis extends work by Joseph Fourier and Norbert Wiener.
Branches include real analysis, complex analysis, harmonic analysis, functional analysis, and measure-theoretic probability, each influenced by figures such as Bernhard Riemann, Augustin-Louis Cauchy, Joseph Fourier, John von Neumann, and Andrey Kolmogorov. Extensions into partial differential equations draw on Sofia Kovalevskaya and Henri Poincaré, while geometric analysis links to Bernhard Riemann's geometry and Élie Cartan's differential systems. Nonstandard analysis arose from later work inspired by Abraham Robinson, and stochastic analysis connects to Kiyoshi Itō's stochastic calculus.
Techniques emphasize epsilon–delta proofs formalized by Karl Weierstrass and sequential methods from Augustin-Louis Cauchy, complemented by measure-theoretic tools of Henri Lebesgue and operator methods of John von Neumann. Complex analytic methods leverage residues and contour integration from Augustin-Louis Cauchy and Bernhard Riemann, while Fourier and spectral techniques trace to Joseph Fourier and David Hilbert. Variational methods and functional inequalities draw on the calculus of variations developed by Leonhard Euler and Joseph-Louis Lagrange, and approximation theory engages contributions from Chebyshev and Andrey Kolmogorov.
Analysis underpins applied and theoretical work across physics and engineering through the calculus of Isaac Newton and Joseph Fourier's heat equation, in quantum mechanics via John von Neumann's operator theory, and in probability and statistics through Andrey Kolmogorov and Ronald Fisher. In signal processing and electrical engineering it uses harmonic analysis from Joseph Fourier and Norbert Wiener, while numerical analysis and computation draw on algorithms shaped by Alan Turing and institutions like Princeton University and Massachusetts Institute of Technology. Economic models and control theory use differential and optimization techniques influenced by Leonid Kantorovich and Richard Bellman.
Standard notation includes the derivative symbols of Gottfried Wilhelm Leibniz and the integral sign of Gottfried Wilhelm Leibniz and Bernhard Riemann, epsilon–delta conventions from Karl Weierstrass, and set and function notation shaped by Georg Cantor and Giuseppe Peano. Functional analytic conventions follow the inner-product and operator frameworks codified by David Hilbert and John von Neumann, while measure-theoretic language uses terminology standardized in part by Henri Lebesgue and later texts from Andrey Kolmogorov.