Elementary theory of analytic functions of one or several by Henri Cartan

By Henri Cartan

Translated from the French, this quantity comprises the substance, with additions, of a process lecture given on the school of technology in Paris.

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It will be an immediate consequence of what follows, to be precise : PROPOSITION lxol < 2. 2 With the conditions of proposition Then the power series p. 2. I, let x0 be such that ( 2. I) has radius of convergence (2. xol· 3 37 POWER SERIES IN ONE VARIABLE Proof of propositio n For r0 < r < ( 2. 3) h 2. 2. Put r0 = lx01, oc. I. r) ! :-oP· q q < h n�O oc. ( n--; p) ! (r-r0)P(r0)•-P ) , n. r• < n�O oo. + ( 2. 1 ) Thus the radius of convergence of the series is > r - r0. x - x01 < p - r0• The double series ( 2.

Reductio ad absurdum supposing that f does not take the The restriction off to concentric circles of centre o defines a continuous deformation of the closed path the integral Definition. £a rZ- Let the origin o. 11 The 1 to a point. Consequently, is zero, which contradicts the hypothesis. 12(t ), where the dot means multiplication of the complex numbers 11(t) and 12(t). The index, with respect to the origin, of the product of two closed paths, which do not pass through o, is equal to the sum of the indices of each of these closed paths.

We shall see later (chapter n, § I, no. 7) what conditions must be satisfied by the open set D for branch oflog t to exist in D. We shall now examine how it is possible to obtain all branches of log t ifone exists. PROPOSITION 5. I If there exists a branch f (t) of log t in the connected open set D, then any other branch is of the for m f(t) + 2k7ri (k an integer); conver sely, f ( t) + 2k7ri is a br anch of log t for arry integer k. 33 POWER SERIES f (t) Let us suppose the that IN ONE VARIABLE g(t) and are two branches of log t.

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