Cauchy and the Creation of Complex Function Theory
In this book, Dr. Smithies analyzes the process through which Cauchy created the basic structure of complex analysis, describing first the eighteenth century background before proceeding to examine the stages of Cauchy's own work, culminating in the proof of the residue theorem and his work on expansions in power series. Smithies describes how Cauchy overcame difficulties including false starts and contradictions brought about by over-ambitious assumptions, as well as the improvements that came about as the subject developed in Cauchy's hands. Controversies associated with the birth of complex function theory are described in detail. Throughout, new light is thrown on Cauchy's thinking during this watershed period. This book is the first to make use of the whole spectrum of available original sources and will be recognized as the authoritative work on the creation of complex function theory.
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1814 memoir Section algebraic analytic function appears applications argument becomes infinite Bulletin de Ferussac Calcul infinitesimal 1823b Cauchy gives Cauchy-Riemann equations Cauchy's theorem Chapter circle closed curves coefficients complex function theory conditions at infinity considers continuous function contributions convergence correction term definite integrals derivatives discussion domain double integrals Euler evaluation of definite Exercices expansion expression finite and continuous finite number formula geometrical language illustrative examples imaginary substitutions infinite series integral residue integrals round integrand introduced Lagrange series Laplace later Legendre limits of integration Maclaurin series method modulus notation obtains Oeuvres 2)2 paper paths plane Poisson polar coordinates polynomials power series principal value proof rational function real and imaginary real variable rectangle remainder remarks residue theorem right-hand side roots second Turin memoir Section 2.3 simple pole supposes tacitly assumed takes usual vanishes writes z)dz zeros