Augustin-Louis Cauchy

Augustin-Louis Cauchy was a French mathematician whose precise definitions and proofs helped establish modern analysis. He also made major contributions to algebra, mechanics, and mathematical physics.

Limit of a sequence: A value that sequence terms approach arbitrarily closely as their indices increase. Cauchy’s definitions of convergence gave limits a central role in analysis.
Cauchy's integral theorem: A theorem stating that the contour integral of a holomorphic function around a suitable closed curve is zero. It is a central result in the complex analysis Cauchy developed.
École Polytechnique: A French engineering and scientific institution founded in 1794. Cauchy studied and later taught at this institution.
Joseph-Louis Lagrange: An Italian-born French mathematician whose work shaped analysis, mechanics, and number theory. Lagrange’s influence on French mathematics formed part of Cauchy’s intellectual inheritance.
Complex analysis: The study of functions of complex variables, especially holomorphic functions and their integrals. Cauchy’s integral results became foundational tools of the subject.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another as their indices increase. Cauchy used this internal closeness condition to characterize convergence.
Cauchy's integral formula: A formula expressing a holomorphic function’s interior values through its values along a surrounding contour. It yields powerful consequences for derivatives and analytic functions.
French Academy of Sciences: France’s principal learned society for the mathematical, physical, and natural sciences. Cauchy became a prominent member and submitted much of his research there.
Pierre-Simon Laplace: A French mathematician and astronomer known for celestial mechanics, probability, and mathematical physics. Laplace belonged to the scientific establishment in which Cauchy trained and worked.
Group theory: The mathematical study of groups, abstract structures describing symmetries and operations. Cauchy’s permutation work helped develop early group theory.

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