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Sequences of Analytic Functions

  • George Pólya
  • Gabor Szegö
Part of the Springer Study Edition book series (SSE)

Abstract

The power series
$${a_1}z + {a_2}{z^2} + \cdots + {a_n}{z^n} + \cdots = \omega $$
which converges not only for z = 0 and for which a 1 ≠ 0 establishes a conformai one to one mapping of a certain neighbourhood of z = 0 onto a certain neighbourhood w = 0. Consequently the relationship between z and w can also be represented by the expansion
$${b_1}\omega + {a_2}{\omega ^2} + \cdots + {b_n}{\omega ^n} + \cdots = z$$
, a 1 b 1 = 1. To compute the second series from the first we set
$$\frac{1}{{{a_1} + {a_2}z + {a_3}{z^2} + \cdots + {a_n}{z^{n - 1}} + \cdots }} = \varphi (z) $$
.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1972

Authors and Affiliations

  • George Pólya
    • 1
  • Gabor Szegö
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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