What was the motivation for Cauchy's Integral Theorem?

Question

How did Cauchy go about Cauchy's integral theorem? What was his motivation?

Answer 1

The original motivation, and an inkling of the integral formula, came from Cauchy's attempts to compute improper real integrals. Here is from A Brief History of Complex Analysis in the 19th Century:

"Cauchy’s first work on complex integration appeared in an 1814 paper on definite integrals (improper real integrals) that was presented to the Institute but not published until 1827 (Bottazzini 1986, 132). In this paper Cauchy describes the method passing from the real to the imaginary realm where one can then calculate an improper integral with ease. This is the first hint of Cauchy’s later famous integral formula and Cauchy-Riemann equations."

The first explicit statement of the theorem dates to Cauchy's 1825 memoir, and is not exactly correct:

"If $f(x + iy)$ is finite and continuous for $x_0\leq x\leq X$, $y_0\leq y\leq Y$, then the value of the integral $\int_{x_0+iy_0}^{X+iY}f(z) dz$ is independent of the form of the functions $x=\phi(t)$ and $y=\psi(t)$."

The proof was not rigorous by modern standards, but it suggests additional motivation, the calculus of variations. After Bernoulli's brachistochrone problem variational problems occupied the best minds of 18th century, including Euler, who also made early contributions to complex analysis. Here is from Cauchy Integral Theorem: a Histortical Development of its Proof by Scott:

"Using the method of calculus of variations, he considered $\phi(t)+\varepsilon u(t)$, $\psi(t)+\varepsilon v(t)$ as an alternate path and showed that the first variation of the integral with respect to $\varepsilon$ vanishes. Note that no mention was made of the continuity of the derivative of $f(z)$ or even of its existence although Cauchy made use of both in his proof. Kline [10] suggests that a possible explanation for this is that Cauchy believed, as others of his time, that a continuous function was always differentiable and that its derivative was discontinuous only where the function itself was discontinuous."

In 1846 Cauchy gave the more familiar proof of the theorem based on Green's formula. It is unclear whether Cauchy knew of Green's work (it was published in a privately printed booklet in 1828, but in a mathematical journal only in 1850), or rediscovered it independently. According to Scott, "there are indications that he was influenced by Green's work because he extended his integral theorem to areas on curved surfaces." In any case, it suggests another motivation, coming from the study of partial differential equations.

A comprehensive account of Cauchy's thought process is Cauchy and the Creation of Complex Function Theory by Smithies.