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A Contrast between complex and real-valued Taylor series

Published online by Cambridge University Press:  09 April 2009

Alexander Abian
Alexander Abian
Affiliation:
Iowa Stare University Ames, Iowa, 50010, U.S.A.
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In this paper it is shown that if c is a point of the region of convergence of an analytic function f(z) = Σ∞ncnZn then in every neighborhood of c there exists a point e such that the value f(c) of the function f(z) is attained by some truncation Σkn=0cnZn off(z) at z = e, i.e., Σkn=0cnen = Σ∞n=0cncn. Also it is shown that the above does not hold in the case of real-valued functions of a real variable.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 4 , December 1974 , pp. 458 - 460
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Saks, S. and Zygmund, A., Analytic Functions (Warsaw 1952).Google Scholar