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In-Class Exercise

Pb. 2.24 By comparing their plots, verify that the above definition for the
Chebyshev polynomial gives the same graph as that obtained from the
closed-form expression:

–1
T (x) = cos(N cos (x)) for 0 ≤ x ≤ 1
N

In addition to the Chebyshev polynomials, you will encounter other
orthogonal polynomials in your engineering studies. In particular, the solu-
tions of a number of problems in electromagnetic theory and in quantum
mechanics (QM) call on the Legendre, Hermite, Laguerre polynomials, etc. In
the following exercises, we explore, in a preliminary manner, some of these
polynomials. We also explore another important type of the special functions:
the spherical Bessel function.

Homework Problems

Pb. 2.25 Plot the function y defined, in each case:
(m + 2 )P ()x = ( m +2 3 )xP () (x − m + 1 )P ()x
m+2 m+1 m
a. Legendre polynomials: 1
() =
Px x and P ()x = ( x −3 2 ) 1
1 2
2
For 0 ≤ x ≤ 1, plot y = P (x)
5
These polynomials describe the electric field distribution from a nonspherical
charge distribution.
 H () xH () m + 1) H ()
x = 2
x − 2(
x
b. Hermite polynomials:  m+2 m+1 2 m
()
 Hx = 2() x and H x = 4 x − 2
1
2
2
m
For 0 ≤ x ≤ 6, plot y = A H x( )exp( −x / 2), where A = ( 2 m! π ) − 1 2/
5 5 m
The function y describes the QM wave-function of the harmonic oscillator.
c. Laguerre polynomials:
 L ( x = 32) [( + m + − x L) ( x −) ( m + 1) 2 L x( )]/( m + 2)
 m+2 m+1 2 m
Lx =−1() x and L x = 1 2( − x + x / 2)
()
 1 2
For 0 ≤ x ≤ 6, plot y = exp(–x/2)L (x)
5
The Laguerre polynomials figure in the solutions of the QM problem of
atoms and molecules.
© 2001 by CRC Press LLC