Question

Solve number 3 with good details. Please do not use other's work.

1. Solve the equation 2. Prove that the series 00 n 2 converges at all points of the unit circle ll-1, except -1 3. Show that there is no convergent power series f(z)- 2, such that f (z) 1 forz 1/2, 1/3, 1/4,... and f(0) 0 l<

Answer 1

If f (z) = sigma^infinity_m = 0 c_nz^n, mod (z) < 2 is a convengent power series with f (z) = 1 at z = 1/2, 1/3, 1/4. .... Then f is analytic on D = mod(z) < 2 and f (z) = 1 at infinitely many points of the Disc with limit in D thus by unioverness theorem we have f (z) = 1 for all z < l implies f (z) = 0 implies f'(0) = 0 we have to choose such for which f'(0) notequalto 0 hence such a function is not possible