Functions of One Complex Variable Special Part

Abstract

Let a ₀, a ₁, a ₂,..., a _n ,... be complex numbers not all zero. Let the power series

$$ f(z) = {a_0} + {a_1}z + {a_2}{z^2} + \cdots + {a_n}{z^n} + \cdots $$

have radius of convergence R, R>0. If R = ∞, f(z) is called an entire function. Let 0 ≦ r < R. Then the sequence

$$ \left| {{a_0}} \right|,\quad \left| {{a_1}} \right|r,\quad \left| {a{}_2} \right|{r^2},\quad \cdots, \;\left| {{a_n}} \right|{r^2},\; \cdots $$

tends to 0, and hence it contains a largest term, the maximum term, whose value is denoted by μ(r). Thus

$$ \left| {{a_n}} \right|{r^n}\underline \leqslant \mu (r) $$

for n = 0, 1, 2, 3,..., r ≧ 0 [I, Ch. 3, § 3].