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b. Find numerically the steady-state solution to this problem using
the techniques of Chapter 4, and assume for some normalized units
the following values for the parameters:

LC = 1, RC = 1, ω = 2π

c. Compare your numerical results with the analytical results.

Application 2: The Legendre Polynomials
We propose to show that the Legendre polynomials are an orthonormal basis
for all functions of compact support over the interval –1 ≤ x ≤ 1. Thus far, we
have encountered the Legendre polynomials twice before. They were defined
through their recursion relations in Pb. 2.25, and in Section 4.7.1 through their
defining ODE. In this application, we define the Legendre polynomials
through their generating function; show how their definitions through their
recursion relation, or through their ODE, can be deduced from their defini-
tion through their generating function; and show that they constitute an
orthonormal basis for functions defined on the interval –1 ≤ x ≤ 1.

1. The generating function for the Legendre polynomials is given by
the simple form:

∞
Gx t(, ) = 1 = ∑ Px t( ) l (7.83)
l
−
12 xt t+ 2
l=0
2. The lowest orders of P (x) can be obtained from the small t-expan-
l
sion of G(x, t); therefore, expanding Eq. (7.83) to first order in t gives:
+
2
2
t
t
1+ xt Ο( ) = P x() + tP x() + Ο( ) (7.84)
0 1
from which, we can deduce that:
Px() = 1 (7.85)
0
Px() = x (7.86)
1

3. By inspection, it is straightforward to verify by substitution that
the generating function satisfies the equation:

+
(12− xt t 2 ∂G + ( −tx ) =G 0 (7.87)
)
∂t

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