# Article

Keywords:
$p$-Laplacian; differential equations in complex domain; extension of $\sin _p$
Summary:
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.
References:
 Burckel, R. B.: An Introduction to Classical Complex Analysis. Vol. 1. Pure and Applied Mathematics 82 Academic Press, New York (1979). MR 0555733 | Zbl 0434.30002
 Pino, M. A. del, Elgueta, M., Manásevich, R.: A homotopic deformation along $p$of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')'+f(t,u)=0$, $u(0)=u(T)=0$, $p>1$. J. Differ. Equations 80 (1989), 1-13. DOI 10.1016/0022-0396(89)90093-4 | MR 1003248
 Elbert, Á.: A half-linear second order differential equation. Qualitative Theory of Differential Equations, Vol. I, Szeged, 1979 Colloq. Math. Soc. János Bolyai 30 North-Holland, Amsterdam (1981), 153-180 M. Farkas. MR 0680591 | Zbl 0511.34006
 Girg, P., Kotrla, L.: Differentiability properties of $p$-trigonometric functions. Electron. J. Differ. Equ. (electronic only) 2014 (2014), 101-127. MR 3344567 | Zbl 1291.33021
 Henrici, P.: Applied and Computational Complex Analysis. Vol. 2: Special Functions-Integral Transforms-Asymptotics-Continued Fractions. Wiley Classics Library John Wiley & Sons, New York (1991). MR 1164865 | Zbl 0925.30003
 Jarník, V.: Differential Equations in the Complex Domain. Academia, Praha Czech (1975). MR 0460758
 Lindqvist, P.: Some remarkable sine and cosine functions. Ric. Mat. 44 (1995), 269-290. MR 1469702 | Zbl 0944.33002
 Markushevich, A. I.: Theory of Functions of a Complex Variable. Three Volumes. Translated and edited by Richard A. Silverman. Chelsea Publishing, New York (1977). MR 0444912
 Wei, D., Liu, Y., Elgindi, M. B.: Some generalized trigonometric sine functions and their applications. Appl. Math. Sci., Ruse 6 (2012), 6053-6068. MR 2981055 | Zbl 1262.42001
 Drexel, The Math Forum @: Discussion: Entire solutions of $f^2+g^2=1$'' of A. Horwitz, P. Vojta, R. Israel, H. P. Boas, B. Dubuque. http://mathforum.org/kb/message.jspa?messageID=21242</b>