Generalized trigonometric functions in complex domain

Girg, Petr, and Kotrla, Lukáš. "Generalized trigonometric functions in complex domain." Mathematica Bohemica 140.2 (2015): 223-239. <http://eudml.org/doc/271566>.

@article{Girg2015,
abstract = {We study extension of $p$-trigonometric functions $\sin _p$ and $\cos _p$ to complex domain. For $p=4, 6, 8, \dots $, the function $\sin _p$ satisfies the initial value problem which is equivalent to (*) \[-(u^\{\prime \})^\{p-2\}u^\{\prime \prime \}-u^\{p-1\} =0, \quad u(0)=0, \quad u^\{\prime \}(0)=1 \] in $\mathbb \{R\}$. In our recent paper, Girg, Kotrla (2014), we showed that $\sin _p(x)$ is a real analytic function for $p=4, 6, 8, \dots $ on $(-\pi _p/2, \pi _p/2)$, where $\pi _p/2 = \int _0^1(1-s^p)^\{-1/p\}$. This allows us to extend $\sin _p$ to complex domain by its Maclaurin series convergent on the disc $\lbrace z\in \mathbb \{C\}\colon |z|<\pi _p/2\rbrace $. The question is whether this extensions $\sin _p(z)$ satisfies (*) in the sense of differential equations in complex domain. This interesting question was posed by Došlý and we show that the answer is affirmative. We also discuss the difficulties concerning the extension of $\sin _p$ to complex domain for $p=3,5,7,\dots $ Moreover, we show that the structure of the complex valued initial value problem (*) does not allow entire solutions for any $p\in \mathbb \{N\}$, $p>2$. Finally, we provide some graphs of real and imaginary parts of $\sin _p(z)$ and suggest some new conjectures.},
author = {Girg, Petr, Kotrla, Lukáš},
journal = {Mathematica Bohemica},
keywords = {$p$-Laplacian; differential equations in complex domain; extension of $\sin _p$},
language = {eng},
number = {2},
pages = {223-239},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized trigonometric functions in complex domain},
url = {http://eudml.org/doc/271566},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Girg, Petr
AU - Kotrla, Lukáš
TI - Generalized trigonometric functions in complex domain
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 2
SP - 223
EP - 239
AB - We study extension of $p$-trigonometric functions $\sin _p$ and $\cos _p$ to complex domain. For $p=4, 6, 8, \dots $, the function $\sin _p$ satisfies the initial value problem which is equivalent to (*) \[-(u^{\prime })^{p-2}u^{\prime \prime }-u^{p-1} =0, \quad u(0)=0, \quad u^{\prime }(0)=1 \] in $\mathbb {R}$. In our recent paper, Girg, Kotrla (2014), we showed that $\sin _p(x)$ is a real analytic function for $p=4, 6, 8, \dots $ on $(-\pi _p/2, \pi _p/2)$, where $\pi _p/2 = \int _0^1(1-s^p)^{-1/p}$. This allows us to extend $\sin _p$ to complex domain by its Maclaurin series convergent on the disc $\lbrace z\in \mathbb {C}\colon |z|<\pi _p/2\rbrace $. The question is whether this extensions $\sin _p(z)$ satisfies (*) in the sense of differential equations in complex domain. This interesting question was posed by Došlý and we show that the answer is affirmative. We also discuss the difficulties concerning the extension of $\sin _p$ to complex domain for $p=3,5,7,\dots $ Moreover, we show that the structure of the complex valued initial value problem (*) does not allow entire solutions for any $p\in \mathbb {N}$, $p>2$. Finally, we provide some graphs of real and imaginary parts of $\sin _p(z)$ and suggest some new conjectures.
LA - eng
KW - $p$-Laplacian; differential equations in complex domain; extension of $\sin _p$
UR - http://eudml.org/doc/271566
ER -