Petr Girg, Lukáš Kotrla, Department of Mathematics and New Technologies for the Information Society, University of West Bohemia, Univerzitní 22, 306 14 Plzeň, Czech Republic, e-mail: pgirg@kma.zcu.cz, kotrla@ntis.zcu.cz
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
$$-(u')^{p-2}u"-u^{p-1} =0, \quad u(0)=0, \quad u'(0)=1 \leqno(*)$$
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 $\{z\in\mathbb{C}\colon|z|<\pi_p/2\}$. 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.
Keywords: $p$-Laplacian; differential equations in complex domain; extension of $\sin_p$
Classification (MSC 2010): 33E30, 34B15, 34M05, 34M99
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