POISSON, Siméon-Denis. Suite du mémoire sur les intégrales définies.
Paris: l'Imprimerie Royal, 1820. First edition.

An important contribution to the foundation of complex function theory. In this paper Possion was the first to carry out integration along a path in the complex plane. "Poisson noted in 1815 and discussed in [this paper] the use of integrals of complex functions taken over paths in the complex plane. As an example he gives ∫dx/x from -1 to 1. Here he sets x=e^iθ, where θ runs from (2n+1)π to 0, and obtaines, by treating the integral as a limit of a sum, the value -(2n+1)πi. Then he notes that the value of an integral need not be the same when taken over an imaginary path as when over a real path. He gives the example ∫(cos ax)/(b²+ x²) dx from -∞ to ∞, where a and b are positive constants. He lets x = t ik, where k is constant and positive, and obtaines the value &

Five memoires in: Journal de l'École Royale Polytechnique, vol. 11. pp. 1-188; 295-341; 342-410; 411-416; 417-489. The complete volume offered [IV,492 pp. and 1 plate - not belonging to the mentioned memoires]. Nice contemporary half calf with gilt spine lettering. Completely clean throughout. A fine copy. ; Louis POINSOT; Augustin-Louis CAUCHY. AMPÈRE, André-Marie; 1. Ampère: Mémoire contenant l'application de la théorie ... à l'intégration des équations aux différentielles partielles du premier et du second ordre; 2. Poisson: Suite du mémoire sur les intégrales définies [KLINE p.633: Poisson was the first man to cary out integration along a path in the complex plane.]; 3. Poinsot: Mémoire sur l'application de l'algèbre à la théorie des nombres; 4. Cauchy: Sur les racines imaginaires des équations; 5. Poisson: Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, et sur l'usage de cette transformation dans la résolution de différens problèmes.