Generalized trigonometric functions in complex domain

Girg, Petr ; Kotrla, Lukáš

Mathematica Bohemica, Tome 140 (2015) no. 2, pp. 223-239.

Résumé

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 (*) $$-(u')^{p-2}u''-u^{p-1} =0, \quad u(0)=0, \quad u'(0)=1 $$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 $\{z\in \mathbb{C}:|z|{\pi }_p/2\}$. 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.

Détail

DOI : 10.21136/MB.2015.144328

Classification : 33E20, 33E30, 34B15, 34M05, 34M99
Mots-clés : $p$-Laplacian; differential equations in complex domain; extension of ${\sin }_p$