Elements of Mathematics Functions of a Real Variable: by Nicolas Bourbaki, Philip Spain (auth.)
By Nicolas Bourbaki, Philip Spain (auth.)
This booklet is an English translation of the final French variation of Bourbaki’s Fonctions d'une Variable Réelle.
The first bankruptcy is dedicated to derivatives, Taylor expansions, the finite increments theorem, convex features. within the moment bankruptcy, primitives and integrals (on arbitrary periods) are studied, in addition to their dependence with appreciate to parameters. Classical services (exponential, logarithmic, round and inverse round) are investigated within the 3rd bankruptcy. The fourth bankruptcy supplies an intensive remedy of differential equations (existence and unicity houses of recommendations, approximate ideas, dependence on parameters) and of platforms of linear differential equations. The neighborhood research of services (comparison kinfolk, asymptotic expansions) is handled in bankruptcy V, with an appendix on Hardy fields. the speculation of generalized Taylor expansions and the Euler-MacLaurin formulation are provided within the 6th bankruptcy, and utilized within the final one to the research of the Gamma functionality at the actual line in addition to at the complicated plane.
Although the subjects of the booklet are almost always of a complicated undergraduate point, they're provided within the generality wanted for extra complex reasons: services allowed to take values in topological vector areas, asymptotic expansions are handled on a filtered set built with a comparability scale, theorems at the dependence on parameters of differential equations are at once appropriate to the examine of flows of vector fields on differential manifolds, etc.