Hardy's generalization of eᶻ and related analogs of cosine and sine
Author(s)
Date
2006Source Title
Computational Methods and Function Theory
Print ISSN
1617-9447
Electronic ISSN
2195-3724
Publisher
Springer
Volume
6
Issue
1
Pages
1 - 14
Language
English
Type
ArticleItem Usage Stats
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Abstract
In 1904, Hardy introduced an entire function depending on two
parameters being a generalization of e z. He had studied in detail its asymptotic
properties and that of its zeros. We consider the two following non-asymptotic
problems related to the zeros. (i) Determine values of the parameters such that
all the zeros belong to the open left half-plane. For these values, the analogs of
sine and cosine generated by Hardy’s function have real, simple and interlacing
zeros. (ii) Determine the number of real zeros as a function of the parameters.
Keywords
Class PIntegral representation
Levin's generalization of the Hermite - Biehler theorem
Logarithmic derivative
Rolle's theorem