Complex Analysis: Theory and Applications · Problem 3.1.5

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · polynomials-complex

Problem

Let

P(z) = a_0 + a_1 z + \cdots + a_n z^n

be a polynomial of degree

n \geq 1

. (a) Show that

P'(z) = a_1 + 2a_2 z + \cdots + na_n z^{n-1}

. (b) Show that, for

k = 0, 1, \ldots, n

a_k = \dfrac{P^{(k)}(0)}{k!}

, where

P^{(k)}

denotes the

k

th derivative of

P

. (By convention,

P^{(0)}(z) = P(z)