nLab Augustin-Louis Cauchy

Augustin-Louis Cauchy was a pioneer in analysis and group theory. He wrote an influential 1821 textbook, Cours d'Analyse.

Cauchy described the basic concepts of differential and integral calculus in terms of limits. His conception of limits was based on infinitesimal variables, which do not appear as such in modern mathematics, although they have been variously identified with sequences (that converge to zero), ultrafilters (that converge to zero), hyperpoints (in the infinitesimal neighbourhood of zero) in the sense of nonstandard analysis, etc. (His infinitesimals were not nilpotent.)

Cauchy's student Karl Weierstrass defined limits in terms of Richard Dedekind's static conceptions of real numbers and functions, thereby creating modern analysis.

Cauchy is associated with:

• Cauchy sequences

• Cauchy sequences (and thus Cauchy nets, Cauchy filters, and Cauchy spaces, although Cauchy himself knew none of these)

• Cauchy–Riemann equations

• Cauchy-Schwarz inequality

• Cauchy integral?

• Cauchy surface

• Cauchy real number

• Cauchy principal value

• Cours d'Analyse, a textbook on infinitesimal analysis/epsilontic analysis

• Cauchy's theorems

  • Cauchy integral theorem and Cauchy integral formula (for contour integrals in complex analysis)

  • Cauchy sum theorem (“Cauchy's mistake”) from his 1821 textbook Cours d'Analyse

  • Cauchy group theorem

category: people