TY - JOUR
T1 - On extremal properties of Jacobian elliptic functions with complex modulus
AU - Siegl, Petr
AU - Štampach, František
N1 - Funding Information:
The research of P.S. is supported by the Swiss National Science Foundation , SNF Ambizione grant No. PZ00P2_154786 . F.Š. gratefully acknowledges the kind hospitality of the Mathematisches Institut, Universität Bern; his research was also supported by grant No. GA13-11058S of the Czech Science Foundation .
Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2016/10/15
Y1 - 2016/10/15
N2 - 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.
AB - 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.
KW - Complex modulus
KW - Extrema of elliptic functions
KW - Jacobian elliptic functions
UR - http://www.scopus.com/inward/record.url?scp=84969542818&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2016.05.008
DO - 10.1016/j.jmaa.2016.05.008
M3 - Article
AN - SCOPUS:84969542818
VL - 442
SP - 627
EP - 641
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 2
ER -