Complex Analysis Suppose f is analytic on a disc Drz0 and un

Complex Analysis

Suppose f is analytic on a disc D_r(z_0) and unbounded (there is no M such that |f(z)| lessthanorequalto M on D_r(z_0)). Then prove that the radius of convergence of the power series expansion of f about z_0 is r.

Solution

If the radius of convergence is , say s >r, then the power series of f(z) about z₀ will converge for all |z|=r.

This implies the boundedness of |f(z) | over the boundary of D_z0 (hence on D_z0) contrary to the assumption.

Thus the radius of convergence is exactly r.