Functions of One Complex Variable Special Part

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


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].


Unit Circle Entire Function Outer Radius Maximum Modulus Equality Sign 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

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

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