Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

The normalisation, whenm=n, is easily found by making the substitution

x=cosθand using (18.55). We immediately obtain

∫ 1

− 1

Tn(x)Tn(x)(1−x^2 )−^1 /^2 dx=

{

π forn=0,

π/2forn=1, 2 , 3 ,.... (18.62)

The orthogonality and normalisation conditions mean that any (reasonable)

functionf(x) can be expanded over the interval|x|<1inaseriesoftheform

f(x)=^12 a 0 +

∑∞

n=1

anTn(x),

where the coefficients in the expansion are given by

an=

2

π

∫ 1

− 1

f(x)Tn(x)(1−x^2 )−^1 /^2 dx.

For the Chebyshev polynomials of the second kind,Un(x), we see from (18.58)

that (1−x^2 )^1 /^2 Un(x)=Vn+1(x) satisfies Chebyshev’s equation (18.54) withν=

n+ 1. Thus, the orthogonality relation for theUn(x), obtained by replacingTi(x)

byVi+1(x) in equation (18.61), reads

∫ 1

− 1

Un(x)Um(x)(1−x^2 )^1 /^2 dx=0 ifn=m.

The corresponding normalisation condition, whenn=m, can again be found by

making the substitutionx=cosθ, as illustrated in the following example.

Show that

I≡

∫ 1

− 1

Un(x)Un(x)(1−x^2 )^1 /^2 dx=

π

2

.

From (18.58), we see that

I=

∫ 1

− 1

Vn+1(x)Vn+1(x)(1−x^2 )−^1 /^2 dx,

which, on substitutingx=cosθ,gives

I=

∫ 0

π

sin(n+1)θsin(n+1)θ

1

sinθ

(−sinθ)dθ=

π

2

.

The above orthogonality and normalisation conditions allow one to expand

any (reasonable) function in the interval|x|<1 in a series of the form

f(x)=

∑∞

n=0

anUn(x),