Cauchy’s foundational research on complex analysis dates roughly from 1814 to 1831. In 1814, he presented his Mémoire sur les intégrales définies (nearly 200 pages long!), a subject that occupied him through to the two Turin memoirs he presented in 1831.
Although he used these results in later research, he made no further substantial contributions to the theory until 1849. Cauchy’s early research is particularly interesting because it belongs to a long tradition of calculations—and questions about whether those calculations were legitimate.
The Cauchy–Riemann equations before their time
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In a 1740 memoir on integration, Alexis Clairaut proved that, when P and Q are polynomials, the differential form \ P(x, y) dx + Q(x, y) dy is exact if the derivative of P with respect to x equals the derivative of Q with respect to y* (a result obtained at the same time by Euler, unbeknownst to Clairaut).
(\ Among other things, a differential form can assign a number to a curve through integration. A differential form is said to be exact* (in Clairaut’s day, the term was "complete") if, in a precise sense, the number associated with a curve depends only on its endpoints.)
