Riemann integral

Riemann integral
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Name	Riemann integral
Introduced	19th century
Inventor	Bernhard Riemann
Field	Analysis

Riemann integral The Riemann integral is a classical construction in mathematical analysis that assigns numbers to certain functions on intervals, developed in the 19th century as part of rigorous calculus. It builds on partitioning intervals and summing function values to capture signed area, and it played a central role in the development of modern analysis alongside work by contemporaries and successors. The notion influenced later formalisms and is linked historically to developments in topology, measure theory, and functional analysis.

Definition

The formal definition uses partitions of a closed interval [a,b] and upper and lower sums to determine integrability, introduced by Bernhard Riemann and clarified by later mathematicians such as Karl Weierstrass, Richard Dedekind, and Augustin-Louis Cauchy. In the standard setup one considers a bounded real-valued function on [a,b] and defines the upper sum and lower sum relative to a partition; the function is integrable if the supremum of lower sums equals the infimum of upper sums, a common value called the integral. This approach links to notions developed by Évariste Galois in contemporaneous mathematical culture and influenced subsequent formalizations by Georg Cantor and Henri Lebesgue. Later expositions by David Hilbert and Emmy Noether situated the definition within broader frameworks used by John von Neumann and Norbert Wiener.

Riemann Sums and Partitions

Riemann sums are formed by choosing partitions and sample points within subintervals and summing products of function values with subinterval widths, a technique appearing in texts by Joseph-Louis Lagrange and Adrien-Marie Legendre as antecedents. The mesh size of a partition controls approximation accuracy, and refinement of partitions leads to convergence of Riemann sums for integrable functions, a theme echoed in lectures by Augustin Cauchy and later in courses by Henri Poincaré. Practical computation of sums and choice of sample points connects historically with problems considered by Carl Friedrich Gauss and Siméon Denis Poisson in numerical quadrature. Influential treatises by Felix Klein and Henri Lebesgue contrasted partition-based Riemann sums with measure-theoretic constructions, while contemporary computational implementations draw on algorithms developed in institutions like Bell Labs and IBM research.

Properties and Criteria for Integrability

Key criteria include boundedness on closed intervals and the set of discontinuities having measure zero, results formalized by Émile Borel and Henri Lebesgue and later presented in texts by Stefan Banach and Maurice Fréchet. Continuous functions on [a,b], treated extensively by Leonhard Euler and Joseph Fourier, are Riemann integrable; functions with only finitely many discontinuities treated by Bernhard Riemann himself are integrable; functions with dense discontinuities may fail to be integrable, as illustrated by examples discussed by Georg Cantor and Richard Dedekind. Monotone functions, examined in the work of Karl Weierstrass, are integrable, and uniform convergence theorems linking sequences of integrable functions to integrability were developed further by David Hilbert and John Littlewood. Counterexamples and pathologies motivating new theories were studied by Émile Borel, Felix Hausdorff, and Nikolai Luzin.

Relationship to Other Integrals

The Riemann integral is related to Lebesgue integration, Daniell integration, and Henstock–Kurzweil integration, with Lebesgue's theory extending integrability to a broader class of functions via measure concepts introduced by Henri Lebesgue and Émile Borel. Daniell's approach, connected to Percy John Daniell, abstracts integration in a way influential for John von Neumann and Norbert Wiener, while the Henstock–Kurzweil integral arose in later 20th-century work linked to Jaroslav Kurzweil. In functional analysis contexts developed by Stefan Banach and John von Neumann, the Riemann integral corresponds to certain linear functionals on spaces of continuous functions, and connections to spectral theory and operator algebras studied by Paul Dirac and John von Neumann appear in advanced treatments. Comparisons with Riemann–Stieltjes integrals, named for Thomas Joannes Stieltjes, show generalizations applicable in probability theory foundational to Andrey Kolmogorov and William Feller.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects antiderivatives and definite integrals, a principle with historical roots in the work of Isaac Newton and Gottfried Wilhelm Leibniz and formal proofs refined by Augustin Cauchy and Karl Weierstrass. The theorem has two parts: one linking differentiation of the integral function to the original integrand under continuity hypotheses, and the other allowing evaluation of the integral via any antiderivative, a viewpoint central to the curricula influenced by Joseph-Louis Lagrange and Adrien-Marie Legendre. Rigorous formulations and counterexamples demonstrating required hypotheses were provided by Émile Picard and Camille Jordan, while applications in differential equations and variational calculus relate to contributions by Leonhard Euler and Joseph-Louis Lagrange. Extensions to vector-valued functions and multivariable settings connect to work by Henri Poincaré and Élie Cartan.

Applications and Examples

Applications span physics, engineering, probability, and geometry, fields advanced by figures such as James Clerk Maxwell, Michael Faraday, Ludwig Boltzmann, and Henri Poincaré. Classical examples include computing areas under polynomials and trigonometric functions studied by Joseph Fourier and Leonhard Euler, and evaluating improper integrals in contexts treated by Augustin Cauchy and Bernhard Riemann. In probability and statistics, the relationship of integrals to expected values features in the foundational work of Thomas Bayes and Ronald Fisher, and numerical quadrature methods trace to pioneers like Carl Friedrich Gauss and Adrien-Marie Legendre. Modern computational implementations draw on numerical analysis developed at institutions such as the Massachusetts Institute of Technology, IBM, and Bell Labs, while advanced theoretical applications link to spectral theory and partial differential equations explored by David Hilbert, John von Neumann, and Olga Ladyzhenskaya.

Category:Mathematical analysis