On extremal properties of Jacobian elliptic functions with complex modulus

Petr Siegl, František Štampach*

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Citation (Scopus)
60 Downloads (Pure)

Abstract

A thorough analysis of values of the function m(mapping)sn( K( m) u| m) for complex parameter m and u∈ (0, 1) is given. First, it is proved that the absolute value of this function never exceeds 1 if m does not belong to the region in C determined by inequalities | z- 1| < 1 and | z| > 1. The global maximum of the function under investigation is shown to be always located in this region. More precisely, it is proved that if u≤ 1/2, then the global maximum is located at m=. 1 with the value equal to 1. While if u> 1/2, then the global maximum is located in the interval (1, 2) and its value exceeds 1. In addition, more subtle extremal properties are studied numerically. Finally, applications in a Laplace-type integral and spectral analysis of some complex Jacobi matrices are presented.

Original languageEnglish
Pages (from-to)627-641
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume442
Issue number2
Early online date10 May 2016
DOIs
Publication statusPublished - 15 Oct 2016
Externally publishedYes

Keywords

  • Complex modulus
  • Extrema of elliptic functions
  • Jacobian elliptic functions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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