Mémoire sur les intégrales définies prises entre des limites imaginaires.

'Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.' 'The shortest path between two truths in the real domain passes through the complex domain' (Jacques Hadamard).

Augustin-Louis Cauchy was one of the greatest mathematicians of the modern era. This pamphlet, of which only 500 were printed, is widely regarded as one of Cauchy's most important contributions to mathematics. With it Cauchy created the key tool for the formalisation of the analysis of complex numbers. The theorem, know known as the Cauchy-Goursat theorem after Goursat managed to remove one of its constraints, allows mathematicians to simplify difficult functions in complex analysis. Technically it says that if two different paths connect the same two points, and a function is holomorphic everywhere 'in between' the two paths, then the two path integrals of the function will be the same.

A note about complex analysis
Complex numbers include the root of -1. They become very useful when trying to solve difficult problems amongst the real numbers - a real number can be represented as the product of two complex numbers. One can then use the theory of complex numbers to deduce a result in the real numbers. Complex analysis is the mathematical field which develops theorems of functions of complex numbers. Whilst to the lay person, complex analysis is arcane, it is incredibly useful for physicists solving problems in such diverse fields as quantum mechanics, fluid dynamics and electrical engineering. A contemporary visual applications is development of complex scenery in computer-generated images; it is possible to generate very complex shapes using complex analysis with little computational power. A good example of this is a digital scene of a field of grass waving in the wind, which can be rendered using the tools of complex analysis. The Mandlebrot Set is another visually arresting (and infinitely complex) illustration of certain complex numbers.