Cauchy

Augustin-Louis Cauchy (1789–1857) was a French mathematician whose work laid foundational principles for analysis and changed how calculus was taught and used. Born in Paris, he pursued mathematics with prolific productivity and held prominent teaching positions at the École Polytechnique and the University of Paris.

In complex analysis, Cauchy introduced the Cauchy-Riemann equations and established the Cauchy integral theorem and Cauchy’s integral formula, providing essential tools for the study of analytic functions. His work helped transform complex analysis into a rigorous and central area of mathematics.

Beyond complex analysis, Cauchy promoted rigorous treatment of limits and infinite processes. He introduced the concept of a Cauchy sequence to formalize convergence and formulated tests for series convergence (the Cauchy criterion). He also contributed to real analysis, numerical methods, and the theory of determinants, and helped popularize ideas about uniform convergence and the interchange of limits.

Several concepts bear his name in various fields: the Cauchy-Schwarz inequality; the Cauchy condensation test for series; the notion of Cauchy sequences; and the Cauchy distribution in probability, a heavy-tailed distribution with density 1/(π(1+x^2)) that has no finite mean or variance; the sum of independent standard Cauchy random variables is again Cauchy.

Cauchy died in Sceaux near Paris in 1857, leaving a lasting legacy in the development of mathematical analysis and its rigorous foundations.