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By Boccardo L., Pellacci B.

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N ( 1 + _£) · n�oo e' = lim r2. J � n n Show that the function of a complex variable z defined by cos z = e;, + 2 e-iz ( . eit resp. sin z = _ e-iz 2i ) is the analytic extension to the whole plane C of the function cos x (resp. sin x) defined in § 3, no. 3, Prove that, for any z, z' e C, cos (z + z') =cos z cos z' - sin z sin z', sin (z + z') =sin z cos z' + cos z sin z'; cos2 z + sin2 z = I. r 3. Prove the relations � x < sin x TC < x for x real and o < x < 7t/2. EXERCISES =x + ry with 14. Let (i) Show that z x, y real.

O 47 POWER (iii ) Let S(X)= }": n�1 SERIES X • n2 IN ONE VARIABLE now and let D be the intersection of the open / disc lzl < 1 and of the open disc lz- I I< constant a such that I. Show that there exists a S(z) + S(1 -z)=a -log zlog (1 ---z) for ze D, where log denotes the principal branch of the complex logarithm in the half�plane Re(�)> o (which contains D). (Note tha�,. if z e D, then log (I -z)= -T(z) with T(X) because of proposition 6. S'(X), of § 3, and that proposition 6. I d = = log ( 1 -z) z � I-Z for 2 of § 3 gives zeD.

C(c + 1) ... (c + n - l) . n - l) x· +. + Show that its sum S(z), for fzJ < p(S), satisfies the differential equation z(1 -z)S" + (c-(a + b + l)z)S' - abS 7. Put Let S(X) s. = a0 + = o. , t. ) · n > === 1. X•. X•, n�O Show that : (i) p(U) = p(V) 1 -- = l-z 8. z•. X• be a formal power series whose coefficients are n�O defined by the. following recurrence relations : a0 = o, a1 = l, a. = 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.

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