Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.4 CHEBYSHEV FUNCTIONS

Sinceδ(Ω−Ω′) can depend only on the angleγbetween the two directions Ω and Ω′,
we may also expand it in terms of a series of Legendre polynomials of the form

δ(Ω−Ω′)=

∑



bP(cosγ). (18.52)

From (18.14), the coefficients in this expansion are given by

b=

2 +1

2

∫ 1

− 1

δ(Ω−Ω′)P(cosγ)d(cosγ)

=

2 +1

4 π

∫ 2 π

0

∫ 1

− 1

δ(Ω−Ω′)P(cosγ)d(cosγ)dψ,

where, in the second equality, we have introduced an additional integration over an
azimuthal angleψabout the direction Ω′(andγis now the polar angle measured from
Ω′to Ω). Since the rest of the integrand does not depend uponψ, this is equivalent
to multiplying it by 2π/ 2 π. However, the resulting double integral now has the form of
a solid-angle integration over the whole sphere. Moreover, when Ω = Ω′, the angleγ
separating the two directions is zero, and so cosγ= 1. Thus, we find

b=

2 +1

4 π

P(1) =

2 +1

4 π

,

and combining this expressionwith (18.51) and (18.52) gives

∑

m

Ym(Ω)Ym∗(Ω′)=

∑



2 +1

4 π

P(cosγ). (18.53)

Comparing this result with (18.49), we see that, to complete the proof of the addition
theorem, we now only need to show that the summations inon either side of (18.53) can
be equatedterm by term.
That such a procedure is valid may be shown by considering an arbitrary rigid rotation
of the coordinate axes, thereby defining new spherical polar coordinatesΩ on the sphere. ̄
Any given spherical harmonicYm(Ω) in the new coordinates can be written as a linear ̄
combination of the spherical harmonicsYm(Ω) of the old coordinates,allhaving thesame
value of. Thus,

Ym(Ω) = ̄

∑

m′=−

Dmm

′

 Y

m′

(Ω),

where the coefficientsDmm′depend on the rotation; note that in this expression Ω andΩ ̄
refer to the same direction, but expressed in the two different coordinate systems. If we
choose the polar axis of the new coordinate system to lie along the Ω′direction, then from
(18.45), withmin that equation set equal to zero, we may write

P(cosγ)=

√

4 π

2 +1

Y^0 (Ω) = ̄

∑

m′=−

C^0 m

′

 Y

m′

(Ω)

for some set of coefficientsC^0 mthat depend on Ω′. Thus, we see that the equality (18.53)
does indeed hold term by term in, thus proving the addition theorem (18.49).

18.4 Chebyshev functions

Chebyshev’s equation has the form

(1−x^2 )y′′−xy′+ν^2 y=0, (18.54)