# IJPAM: Volume 37, No. 4 (2007)

**A NOTE ON THE SECOND SOLUTION**

OF CHEBYSHEV'S EQUATION

OF CHEBYSHEV'S EQUATION

Departamento de Investigación en Física

Universidad de Sonora

Apdo. Postal 5-088, Hermosillo, Son., 83190, MÉXICO

e-mail: [email protected]

e-mail: [email protected]

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 ,
their second solution is very different in nature, namely, in the case of
Legendre equation the functions have a singularity at while
Chebyshev ones are well behaved in all the interval. Regarding the second
solution in , the situation is more dramatic since 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 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, and , 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