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Generalized trigonometric functions in complex domain

We study extension of p-trigonometric functions sinp and cosp to complex domain. For p = 4, 6, 8, . . ., the function sinp satisfies the initial value problem which is equivalent to (∗) −(u ′ ) p−2 u ′′ − u p−1 = 0, u(0) = 0, u ′ (0) = 1 in R. In our recent paper, Girg, Kotrla (2014), we showed that sinp(x) is a real analytic function for p = 4, 6, 8, . . . on (−πp/2, πp/2), where πp/2 = R 1 0 (1 …