Question

1 - 22) sin(Tz) Consider fe)() a. Find all isolated singularities of f in C and classify each as removable, a pole (specity the order), or essential. b. Explain, with reference to part (a), why f has a series expansion of the form Σ000 ch3k valid near 0. c. Find co d. What is the radius of convergence of the series in part (b)?

Answer 1

f (z) = (1 - z^2) sin (pi z)/z cos (pi z/2) a) The function f has singularities at z , if z_0 cos (pi z/2) = 0 Therefore the singularities of f are 0 plusminus 1, plusminus 3, plusminus 5 . . . plusminus (2n + 1) n elementof N clearly each of those points are isolated, therefore those singularities are isolated singularities Now, lim_z Rightarrow 0 (1 - z^2) sin (pi z)/z cos (pi z/2) = lim_z Rightarrow 0 pi cos (pi z) (1 - z^2) - 2z sin pi z/cos pi z/2 - pi/2 z sin (pi z/2) = pi/1 = pi That means lim_z Rightarrow 0 f (z) exists finitely, so 0 is a removable singularities Now, llim_z Rightarrow 2n + 1 (1 - z^2) sin (pi z)/z cos (pi z/2) = lim_z Rightarrow 2n + 1 pi (1 - z^2) cos pi z - 2z sin (pi z)/cos (pi z/2) - pi/2 z sin (pi z/2)