By Henri Cartan

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

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**Example text**

These two theorems are consequences of a lemma which we shall now state. First of all, here is a definition : Definition. Let (t, u) - S(t, u) be a continuous mapping of a rectangle (6. 3) a'< u< b' into the open set D, and let following the mapping S is a w be a closed form in D. continuous function f(t, u) primitive of A w in the rectangle satisfying the following condition : (P') For any point (-r, u ) of the rectangle, these exists a primitive F of w in a neigh bourhoodof o(-r, u ) such that F(o(t, u)) = f(t, u) at any point (t, u) s�fficiently near to (-r, u ) .

Oi:an-1 + �an-2 for n > 2, where a, � are given real numbers. a) Show that, for n > 1, we have fa . , l�I, 1/2) and deduce that the radius of convergence p(S) =F o. b) Show that (1 -az- �z2)S(z) = z, for lzl < p(S), and deduce that, for fzl < p(S), S(z) = c) z l -az - �z2 · Let z1, z2 be the two roots of �X2 +aX - I = o. By decomposing 45 POWER SERIES IN ONE VARIABLE the right hand side of (1) into partial fractions, find an expression for the an in terms of z1 and z2 and deduce that p(S) =min

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. 3). Its sum can therefore be calculated by regrouping the terms in an arbitrary manner. this sum in two different ways. h a. n! h (O�p�nP· (n 1 r (p + q) ! ;-0 Since > p - r0• 1 ) p. ;-0 = S(x); another grouping gives Formula ( 2. 2) follows from a comparison of these two and this completes the proof. Note I. than p The radius of convergence of series -lxol· S(X) Then S(x) I ix -- I - = = ( 2.

### Elementary theory of analytic functions of one or several complex variables by Henri Cartan

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