417 research outputs found
The violation of a uniqueness theorem and an invariant in the application of Poincar\'{e}--Perron theorem to Heun's equation
The domain of convergence of a Heun function obtained through the
Poincar\'{e}--Perron (P--P) theorem is not absolute convergence but conditional
one [2]. We show that a uniqueness theorem is not available if we apply the
P--P theorem into the Heun's equation. We verify that the uniqueness theorem is
only applicable when a local Heun function is absolutely convergent.Comment: 10 pages, 5 figures. Change the title and abstract. The number of
pages has been reduced. Correct some of the problems with Englis
The radius of convergence of the Heun function
Heun functions generalize well-known special functions such as Spheroidal
Wave, Lame, Mathieu, and hypergeometric--type functions. They are applicable to
diverse areas such as theory of black holes, lattice systems in statistical
mechanics, solutions of the Schrodinger equation of quantum mechanics, and
addition of three quantum spins.
We consider the radius of convergence of the Heun function, and we show why
the Poincare-Perron (P-P) theorem is not available for the absolute convergence
since it is applied to the Heun's equation. Moreover, we construct the absolute
convergence test in which is suitable for the three term recurrence relation in
a power series.Comment: 14 pages. 3 figures. The number of pages has been reduced. Give the
reader greater insight into what the key results are in the introduction.
Correct some of the problems with Englis
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