Download PHYS 301 -- Introduction to Mathematical Physics Chapter 12 Series

PHYS 301 -- Introduction to Mathematical Physics
Chapter 12 Series Solutions of Differential Equations…(Section 1-4 Legendre Polynomials)
(i)
For a second order linear ODE that is nonlinear with non-constant coefficients, we cannot
use methods in Chapter 8 directly. However, if a solution of it can be expressed into a power
series of x, it is possible to find it by substituting the unknown series into the ODE and solve for
relationships between coefficients of different terms.
(ii)
As an example, we consider the Legendre differential equation, which is frequently
encountered in solving partial differential equations using spherical coordinates.
(iii) If l is a non-negative integral in the Legendre differential equation, one series solution of
it only has terms up to x l , and so it is called Legendre polynomials Pl (x) , defined with Pl (1) = 1.
(iv)
Pl (x) can also be found using the Gram-Schmidt Method (pp 182-183) by requiring
1
1
€ = 0 if l ≠ m . In fact, it can also be shown€
that ∫ x
∫ P (x)P (x)dx
l
m
−1
m
Pl (x)dx = €
0 if m < l . Also,
−1
€ Pl (x) is an even function with respect to x for even l, and is an odd function for odd l.
(v)
€
€
€
1 €
dl 2
Pl (x) = l
(x −1) l .
Pl (x) can also be expressed in Rodrigues’ formula:
€
l
2 l! dx
Key equations:
€
(12.2.1)
Legendre differential equation.
€
(12.2.7)
Series Solutions of the Legendre differential equation.
(12.2.8)
The first few Legendre Polynomials.
(12.4.1)
Legendre Polynomials expressed in Rodrigues’ formula.
Examples: 12.1.2, 12.1.3, 12.2.2, 12.3.6, 12.4.4