Astroparticle Physics

386 A Mathematical Appendix‘periodic functions’important differentialequationsnabla operatorseparation of variablesWhen, however, a function to describe the measuredtemperature as a function of direction is chosen, one cannottake a simple polynomial in θ and φ, because this wouldnot satisfy the obvious continuity requirements, e.g., that thefunction at φ = 0 matches that at φ = 2π. By using sphericalharmonics as the basis functions for the expansion, theserequirements are automatically taken into account.Now one has to remember how the spherical harmonicsare defined. Several important differential equations ofmathematical physics (Schrödinger, Helmholtz, Laplace) canbe written in the form( )∇ 2 + v(r) ψ = 0 ,(A.1)where ∇ is the usual nabla operator, as defined by∂∇ = e x∂x + e ∂y∂y + e ∂z∂z .(A.2)Here v(r) is an arbitrary function depending only on the radialcoordinate r. In separation of variables in spherical coordinates,a solution of the formψ(r,θ,φ) = R(r) Θ(θ) Φ(φ)(A.3)angular partsis tried. Substituting this back into (A.1) gives for the angularpartsd 2 Φdφ 2 =−m2 Φ,d 2 Θdθ 2 + cos θ[]dΘsin θ dθ + l(l + 1) −m2sin 2 Θ = 0 ,θ(A.4)azimuthal solutionThe solution for Θ is proportional to the associated Leg-endre function Plm (cos θ). The product of the two angularparts is called the spherical harmonic function Y lm (θ, φ),polar solutionspherical harmonic functionwhere l = 0, 1,...and m =−l,...,l are separation constants.The solution for Φ isΦ(φ) = 1 √2πe imφ .(A.5)Y lm (θ, φ) = Θ(θ)Φ(φ)√2l + 1 (l − m)!=4π (l + m)! P l m (cos θ)e imφ . (A.6)