IJPAM: Volume 37, No. 4 (2007)


J.L. Marín$^1$, I.A. Marín-Enriquez$^2$, R. Pérez-Enriquez$^3$
$^{1,2}$Departamento de Investigación en Física
Universidad de Sonora
Apdo. Postal 5-088, Hermosillo, Son., 83190, MÉXICO
$^1$e-mail: [email protected]
$^2$e-mail: [email protected]
$^3$Departamento de Física
Universidad de Sonora
Apdo. Postal 1626, Hermosillo, Son., 83000, MÉXICO
e-mail: [email protected]

Abstract.In spite of that Chebyshev equation is very similar to Legendre equation, in the sense that their first solution span an orthogonal basis in $[-1,1]$, their second solution is very different in nature, namely, in the case of Legendre equation the functions $Q_n$ have a singularity at $\pm 1$ while Chebyshev ones are well behaved in all the interval. Regarding the second solution in $[1,\infty )$, the situation is more dramatic since $Q_n$ are still singular at 1 and goes to zero at infinity, while Chebyshev second solution is well behaved at 1 but diverges at infinity. However, certain physical applications demand that Chebyshev equation second solution behaves as $Q_n$ when the argument is large. In such a case, the only possibility to get a second solution of the equation consists in finding a Frobenius series representation. In this work we discuss the properties of the second solution of Chebyshev equation in both, $[-1,1]$ and $[1,\infty )$, a matter that, to our knowledge has not been discussed nor in textbooks or current literature.

Received: March 3, 2007

AMS Subject Classification: 33E99, 34A05, 34A25

Key Words and Phrases: Chebyshev equation, elliptic coordinates, second solution, Frobenius method

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 37
Issue: 4