Mathematical methods Lecture 19 of 34

October 19, 2012 by Multimedia Publications and Printing Services

Lecturer: K.S. Narain , ICTP

This lesson continues with the study of the generalized Rodrigues' polynomials recalling some of the results of the previous lesson and continues with the study of the differential equations that are satisfied by these polynomials. It was shown that the fallowing equation holds 1/W ?/?x(SW ?/?x Cn)=-?nCn. As an example of the validity of this equation the Hermite polynomials were considered, it was shown that any second degree polynomial n can be written as a linear combination of  Hermite polynomials to order 2 (H0,H1,H2).

From the previous differential equation and using the properties of the polynomials the constant an was calculated that was the only unknown in that expression. For the case of the Hermite polynomials the differential equation becomes ?2/?x2 Hn -2x?/?x Hn + 2n Hn = 0, for the Laguerre polynomials  it is x ?2/?x2 Lvn + (-x+v+1) ?/?x  Lvn + n Lvn = 0 and for and for the Legendre polynomials we have (1-x2) ?2/?x2 Pn + (-2x) ?/?x Pn   -(-2n-n(n-1)+n(n+1)) Pn = 0.

Mathematics for Physicists (Text Book from Google e-books preview)

Written by Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4

Using the general differential equation for the Rodrigues' polynomials derived in this lesson calculate the differential equation that the Jacobi polynomials satisfied.

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