Petr Girg, Lukas Kotrla
Abstract:
p-trigonometric functions are generalizations of the trigonometric functions.
They appear in context of nonlinear differential equations
and also in analytical geometry of the p-circle in the plain.
The most important p-trigonometric function is
.
For p>1, this function is defined as the unique solution
of the initial-value 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: p-Laplacian; p-trigonometry; 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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