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$p$-Laplacian; differential equations in complex domain; extension of $\sin _p$
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 $\{z\in \mathbb {C}\colon |z|<\pi _p/2\}$. 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.
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