Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

288

Gram–Schmidt process

Problem 202 Determine the first three terms of Legendre polynomials by the Gram–Schmidt process, if the standard base { 1 , x , x 2 } is given.

Solution Legendre polynomials must fulfill the orthogonality condition 1 Z − 1 p ( x ) q ( x ) d x = 0 . If the first term is y 1 = 1, then

1 R − 1 1 R − 1

x · 1d x

y 2 = x −

= x ,

1 · 1d x

1 R − 1

1 R − 1

x 2 · 1d x

x 2 · 1d x

1 3

y 3 = x 2 −

1 −

x = x 2 −

.

1 R − 1

1 R − 1

1 · 1d x

x · x d x

Theorem18 If the Sturm–Liouville problem (5.229), (5.230) satisfies the conditions of the previous theorem, and if p̸ = 0 in the entire interval a ≤ x ≤ b , then the main values of the problem are real.

Solution Let us assume the opposite, namely, that λ = α + i β is a main value of the problem, and the respective main function has the form y ( x )= u ( x )+ iv ( x ) . In these expressions α , β are real constants, and u and v are real functions. By substituting these values into equation (5.229) we obtain ru ′ + irv ′ ′ +( q + α p + i β p )( u + iv )= 0 . In order for this complex equation to be satisfied, it is necessary that both its real and imaginary parts be equal to zero, i.e. ru ′ ′ +( q + α p ) u − β pv = 0 , rv ′ ′ +( q + α p ) v + β pu = 0 .