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Generalized trigonometric functions in complex domain

Petr Girg, Lukáš Kotrla (2015)

Mathematica Bohemica

We study extension of p -trigonometric functions sin p and cos p to complex domain. For p = 4 , 6 , 8 , , the function sin p satisfies the initial value problem which is equivalent to (*) - ( u ' ) p - 2 u ' ' - u p - 1 = 0 , u ( 0 ) = 0 , u ' ( 0 ) = 1 in . In our recent paper, Girg, Kotrla (2014), we showed that sin p ( x ) is a real analytic function for p = 4 , 6 , 8 , on ( - π p / 2 , π p / 2 ) , where π p / 2 = 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 : | z | < π p / 2 } . The question is whether this extensions sin p ( z ) satisfies (*) in the sense of differential equations in complex domain. This interesting...

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