Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

266

Convergence with respect to norm is also called mean-square convergence or meancon vergence . According to this definition the series (5.227) converges (with respect to norm) to function f if

b Z a

2 dx

lim n → ∞

[ s n ( x ) − f ( x )]

= 0 ,

where s n ( x ) is the partial sum of the series (5.227)

n ∑ k = 1

s n ( x )=

a k g k ( x ) .

Definition The set of orthonormal functions g 1 , g 2 ,... is complete in the set of functions S on the interval a ≤ x ≤ b , if any function f from S can be approximated with arbitrary accuracy by the linear combination a 1 g 1 + a 2 g 2 + ··· + a n g n . This means that for each ε > 0 constants a 1 , a 2 ,..., a n can be found, such that for a sufficiently large n ∥ f − ( a 1 g 1 + a 2 g 2 + ... + a n g n ) ∥ < ε .

It can be shown that the sets of Legendre polynomials and Bessel functions are complete in the set of continuous real functions on appropriate intervals.

5.9.3 Sturm–Liouville problem

In engineering, various important orthogonal function sets appear as solutions of second-order linear differential equations of the form r ( x ) y ′ ′ +[ q ( x )+ λ p ( x )] y = 0 , (5.229) on an interval a ≤ x ≤ b , with the following boundary conditions Here λ is a parameter and k i , that is, l i ( i = 1 , 2), are given (known) real constants, which are not both equal to zero. The equation (5.229) is called the Sturm 17 – Liouville 18 equation . It can be shown that Legendre, Bessel and some other equations can be represented in this form. The problem of solving the differential equation (5.229) with boundary conditions (5.230) is called the Sturm–Liouville problem . 17 Jacques Charles Francois Sturm (1803-1855), French mathematician of Swiss origin. He made a significant contribution to algebra, and is known for being the first to calculate the speed of sound in water. 18 Joseph Liouville (1809-1882), French mathematician. He made important contributions in various fields of mathematics, and his work in complex analysis, special functions, differential geometry and number theory is especially well known. a ) k 1 y ( a )+ k 2 y ′ ( a )= 0 , b ) l 1 y ( b )+ l 2 y ′ ( b )= 0 . (5.230)