Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 101127.
Differentiability properties of ptrigonometric functions
Petr Girg, Lukas Kotrla
Abstract:
ptrigonometric functions are generalizations of the trigonometric functions.
They appear in context of nonlinear differential equations
and also in analytical geometry of the pcircle in the plain.
The most important ptrigonometric function is
.
For p>1, this function is defined as the unique solution
of the initialvalue problem
for any
.
We prove that the
th derivative of
can be expressed in the form
on
, where
,
and
.
Using this formula, we proved the order of differentiability
of the function
.
The most surprising (least expected)
result is that
if p is an even integer.
This result was essentially used in the proof of theorem, which says that the
Maclaurin series of
converges on
if
p is an even integer. This completes previous results that were
known e.g. by Lindqvist and Peetre where this convergence was conjectured.
Published February 10, 2014.
Math Subject Classifications: 34L10, 33E30, 33F05.
Key Words: pLaplacian; ptrigonometry; analytic functions; approximation.
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Petr Girg
Department of Mathematics, University of West Bohemia
Univerzitni 22, 30614
Plzen, Czech Republic
email: pgirg@kma.zcu.cz


Lukas Kotrla
Department of Mathematics, University of West Bohemia
Univerzitni 22, 30614
Plzen, Czech Republic
email: kotrla@students.zcu.cz

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