Complex Analysis: Theory and Applications · Problem 1.2.8

Complex Numbers — Howell & Matthews

Mathematics — Complex Analysis · binomial-theorem-complex

Problem

Use mathematical induction to show that the binomial theorem is valid for complex numbers. In other words, show that, if

z

and

w

are arbitrary complex numbers and

n

is a positive integer, then

(z + w)^n = \sum_{k=0}^{n} \binom{n}{k} z^k w^{n-k}, \quad \text{where} \quad \binom{n}{k} = \frac{n!}{k!(n-k)!}.