| Abstrakt: |
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. |
