Mathematical Methods for Physics and Engineering - Matematica.NET

18.4 CHEBYSHEV FUNCTIONSSince δ(Ω − Ω ′ ) can depend only on the angle γ between the two directions Ω and Ω ′ ,we may also expand it in terms of a series of Legendre polynomials of the formδ(Ω − Ω ′ )= ∑ lb l P l (cos γ). (18.52)From (18.14), the coefficients in this expansion are given byb l ==2l +122l +14π∫ 1δ(Ω − Ω ′ )P l (cos γ) d(cos γ)−1∫ 2π ∫ 10−1δ(Ω − Ω ′ )P l (cos γ) d(cos γ) dψ,where, in the second equality, we have introduced an additional integration over anazimuthal angle ψ about the direction Ω ′ (and γ is now the polar angle measured fromΩ ′ to Ω). Since the rest of the integrand does not depend upon ψ, this is equivalentto multiplying it by 2π/2π. However, the resulting double integral now has the form ofa solid-angle integration over the whole sphere. Moreover, when Ω = Ω ′ , the angle γseparating the two directions is zero, and so cos γ = 1. Thus, we find2l +1b l =4π P 2l +1l(1) =4π ,and combining this expression with (18.51) and (18.52) gives∑Yl m (Ω)Ylm∗ (Ω ′ )= ∑ 2l +14π P l(cos γ). (18.53)lmlComparing this result with (18.49), we see that, to complete the proof of the additiontheorem, we now only need to show that the summations in l on either side of (18.53) canbe equated term by term.That such a procedure is valid may be shown by considering an arbitrary rigid rotationof the coordinate axes, thereby defining new spherical polar coordinates ¯Ω on the sphere.Any given spherical harmonic Yl m(¯Ω)in the new coordinates can be written as a linearcombination of the spherical harmonics Yl m (Ω) of the old coordinates, all having the samevalue of l. Thus,l∑Yl m (¯Ω) = D mm′ (Ω),m ′ =−ll Ylm′where the coefficients Dlmm′ depend on the rotation; note that in this expression Ω and ¯Ωrefer to the same direction, but expressed in the two different coordinate systems. If wechoose the polar axis of the new coordinate system to lie along the Ω ′ direction, then from(18.45), with m in that equation set equal to zero, we may write√4πl∑P l (cos γ) =2l +1 Y l 0 (¯Ω) = Cl 0m′ Ylm′ (Ω)m ′ =−lfor some set of coefficients Cl 0m that depend on Ω ′ . Thus, we see that the equality (18.53)does indeed hold term by term in l, thus proving the addition theorem (18.49). ◭18.4 Chebyshev functionsChebyshev’s equation has the form(1 − x 2 )y ′′ − xy ′ + ν 2 y =0, (18.54)595