# Functions of One Complex Variable Special Part

• George Pólya
• Gabor Szegö
Part of the Classics in Mathematics book series (CLASSICS)

## 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].

## Keywords

Unit Circle Entire Function Outer Radius Maximum Modulus Equality Sign
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• George Pólya
• 1
• Gabor Szegö
• 1
1. 1.Stanford UniversityStanfordUSA

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