Appendix E Properties of special functions E.1 Bessel functions Notation z = complex number; ν,x = real numbers; n = integer J ν (z) = ordinary Bessel function of the ?rst kind N ν (z) = ordinary Bessel function of the second kind I ν (z) = modi?ed Bessel function of the ?rst kind K ν (z) = modi?ed Bessel function of the second kind H (1) ν = Hankel function of the ?rst kind H (2) ν = Hankel function of the second kind j n (z) = ordinary spherical Bessel function of the ?rst kind n n (z) = ordinary spherical Bessel function of the second kind h (1) n (z) = spherical Hankel function of the ?rst kind h (2) n (z) = spherical Hankel function of the second kind f prime (z) = df(z)/dz = derivative with respect to argument Di?erentialequations d 2 Z ν (z) dz 2 + 1 z dZ ν (z) dz + parenleftbigg 1 ? ν 2 z 2 parenrightbigg Z ν (z) = 0 (E.1) Z ν (z) = ? ? ? ? ? ? ? J ν (z) N ν (z) H (1) ν (z) H (2) ν (z) (E.2) N ν (z) = cos(νπ)J ν (z)? J ?ν (z) sin(νπ) ,νnegationslash= n, |arg(z)| <π (E.3) H (1) ν (z) = J ν (z)+ jN ν (z) (E.4) H (2) ν (z) = J ν (z)? jN ν (z) (E.5) d 2 ˉ Z ν (x) dz 2 + 1 z d ˉ Z ν (z) dz ? parenleftbigg 1 + ν 2 z 2 parenrightbigg ˉ Z ν = 0 (E.6) ˉ Z ν (z) = braceleftbigg I ν (z) K ν (z) (E.7) L(z) = braceleftbigg I ν (z) e jνπ K ν (z) (E.8) I ν (z) = e ?jνπ/2 J ν (ze jπ/2 ), ?π<arg(z) ≤ π 2 (E.9) I ν (z) = e j3νπ/2 J ν (ze ?j3π/2 ), π 2 < arg(z) ≤ π (E.10) K ν (z) = jπ 2 e jνπ/2 H (1) ν (ze jπ/2 ), ?π<arg(z) ≤ π 2 (E.11) K ν (z) =? jπ 2 e ?jνπ/2 H (2) ν (ze ?jπ/2 ), ? π 2 < arg(z) ≤ π (E.12) I n (x) = j ?n J n (jx) (E.13) K n (x) = π 2 j n+1 H (1) n (jx) (E.14) d 2 z n (z) dz 2 + 2 z dz n (z) dz + bracketleftbigg 1 ? n(n + 1) z 2 bracketrightbigg z n (z) = 0, n = 0,±1,±2,... (E.15) z n (z) = ? ? ? ? ? ? ? j n (z) n n (z) h (1) n (z) h (2) n (z) (E.16) j n (z) = radicalbigg π 2z J n+ 1 2 (z) (E.17) n n (z) = radicalbigg π 2z N n+ 1 2 (z) (E.18) h (1) n (z) = radicalbigg π 2z H (1) n+ 1 2 (z) = j n (z)+ jn n (z) (E.19) h (2) n (z) = radicalbigg π 2z H (2) n+ 1 2 (z) = j n (z)? jn n (z) (E.20) n n (z) = (?1) n+1 j ?(n+1) (z) (E.21) Orthogonalityrelationships integraldisplay a 0 J ν parenleftBig p νm a ρ parenrightBig J ν parenleftBig p νn a ρ parenrightBig ρ dρ = δ mn a 2 2 J 2 ν+1 (p νn ) = δ mn a 2 2 bracketleftbig J prime ν (p νn ) bracketrightbig 2 ,ν>?1 (E.22) integraldisplay a 0 J ν parenleftbigg p prime νm a ρ parenrightbigg J ν parenleftbigg p prime νn a ρ parenrightbigg ρ dρ = δ mn a 2 2 parenleftbigg 1 ? ν 2 p prime2 νm parenrightbigg J 2 ν (p prime νm ), ν > ?1 (E.23) integraldisplay ∞ 0 J ν (αx)J ν (βx)xdx= 1 α δ(α?β) (E.24) integraldisplay a 0 j l parenleftBig α lm a r parenrightBig j l parenleftBig α ln a r parenrightBig r 2 dr = δ mn a 3 2 j 2 n+1 (α ln a) (E.25) integraldisplay ∞ ?∞ j m (x)j n (x) dx = δ mn π 2n + 1 , m, n ≥ 0 (E.26) J m (p mn ) = 0 (E.27) J prime m (p prime mn ) = 0 (E.28) j m (α mn ) = 0 (E.29) j prime m (α prime mn ) = 0 (E.30) Speci?cexamples j 0 (z) = sin z z (E.31) n 0 (z) =? cos z z (E.32) h (1) 0 (z) =? j z e jz (E.33) h (2) 0 (z) = j z e ?jz (E.34) j 1 (z) = sin z z 2 ? cos z z (E.35) n 1 (z) =? cos z z 2 ? sin z z (E.36) j 2 (z) = parenleftbigg 3 z 3 ? 1 z parenrightbigg sin z ? 3 z 2 cos z (E.37) n 2 (z) = parenleftbigg ? 3 z 3 + 1 z parenrightbigg cos z ? 3 z 2 sin z (E.38) Functionalrelationships J n (?z) = (?1) n J n (z) (E.39) I n (?z) = (?1) n I n (z) (E.40) j n (?z) = (?1) n j n (z) (E.41) n n (?z) = (?1) n+1 n n (z) (E.42) J ?n (z) = (?1) n J n (z) (E.43) N ?n (z) = (?1) n N n (z) (E.44) I ?n (z) = I n (z) (E.45) K ?n (z) = K n (z) (E.46) j ?n (z) = (?1) n n n?1 (z), n > 0 (E.47) Powerseries J n (z) = ∞ summationdisplay k=0 (?1) k (z/2) n+2k k!(n + k)! (E.48) I n (z) = ∞ summationdisplay k=0 (z/2) n+2k k!(n + k)! (E.49) Smallargumentapproximations |z|lessmuch1. J n (z) ≈ 1 n! parenleftBig z 2 parenrightBig n (E.50) J ν (z) ≈ 1 Gamma1(ν+ 1) parenleftBig z 2 parenrightBig ν (E.51) N 0 (z) ≈ 2 π (ln z + 0.5772157 ? ln 2) (E.52) N n (z) ≈? (n ? 1)! π parenleftbigg 2 z parenrightbigg n , n > 0 (E.53) N ν (z) ≈? Gamma1(ν) π parenleftbigg 2 z parenrightbigg ν ,ν>0 (E.54) I n (z) ≈ 1 n! parenleftBig z 2 parenrightBig n (E.55) I ν (z) ≈ 1 Gamma1(ν+ 1) parenleftBig z 2 parenrightBig ν (E.56) j n (z) ≈ 2 n n! (2n + 1)! z n (E.57) n n (z) ≈? (2n)! 2 n n! z ?(n+1) (E.58) Largeargumentapproximations |z|greatermuch1. J ν (z) ≈ radicalbigg 2 πz cos parenleftBig z ? π 4 ? νπ 2 parenrightBig , |arg(z)| <π (E.59) N ν (z) ≈ radicalbigg 2 πz sin parenleftBig z ? π 4 ? νπ 2 parenrightBig , |arg(z)| <π (E.60) H (1) ν (z) ≈ radicalbigg 2 πz e j(z? π 4 ? νπ 2 ) , ?π<arg(z)<2π (E.61) H (2) ν (z) ≈ radicalbigg 2 πz e ?j(z? π 4 ? νπ 2 ) , ?2π<arg(z)<π (E.62) I ν (z) ≈ radicalbigg 1 2πz e z , |arg(z)| < π 2 (E.63) K ν (z) ≈ radicalbigg π 2z e ?z , |arg(z)| < 3π 2 (E.64) j n (z) ≈ 1 z sin parenleftBig z ? nπ 2 parenrightBig , |arg(z)| <π (E.65) n n (z) ≈? 1 z cos parenleftBig z ? nπ 2 parenrightBig , |arg(z)| <π (E.66) h (1) n (z) ≈ (?j) n+1 e jz z , ?π<arg(z)<2π (E.67) h (2) n (z) ≈ j n+1 e ?jz z , ?2π<arg(z)<π (E.68) Recursionrelationships zZ ν?1 (z)+ zZ ν+1 (z) = 2νZ ν (z) (E.69) Z ν?1 (z)? Z ν+1 (z) = 2Z prime ν (z) (E.70) zZ prime ν (z)+νZ ν (z) = zZ ν?1 (z) (E.71) zZ prime ν (z)?νZ ν (z) =?zZ ν+1 (z) (E.72) zL ν?1 (z)? zL ν+1 (z) = 2νL ν (z) (E.73) L ν?1 (z)+ L ν+1 (z) = 2L prime ν (z) (E.74) zL prime ν (z)+νL ν (z) = zL ν?1 (z) (E.75) zL prime ν (z)?νL ν (z) = zL ν+1 (z) (E.76) zz n?1 (z)+ zz n+1 (z) = (2n + 1)z n (z) (E.77) nz n?1 (z)?(n + 1)z n+1 (z) = (2n + 1)z prime n (z) (E.78) zz prime n (z)+(n + 1)z n (z) = zz n?1 (z) (E.79) ?zz prime n (z)+ nz n (z) = zz n+1 (z) (E.80) Integralrepresentations J n (z) = 1 2π integraldisplay π ?π e ?jnθ+jzsin θ dθ (E.81) J n (z) = 1 π integraldisplay π 0 cos(nθ ? z sin θ)dθ (E.82) J n (z) = 1 2π j ?n integraldisplay π ?π e jzcos θ cos(nθ)dθ (E.83) I n (z) = 1 π integraldisplay π 0 e z cos θ cos(nθ)dθ (E.84) K n (z) = integraldisplay ∞ 0 e ?z cosh(t) cosh(nt) dt, |arg(z)| < π 2 (E.85) j n (z) = z n 2 n+1 n! integraldisplay π 0 cos(z cos θ)sin 2n+1 θ dθ (E.86) j n (z) = (?j) n 2 integraldisplay π 0 e jzcos θ P n (cos θ)sin θ dθ (E.87) Wronskiansandcrossproducts J ν (z)N ν+1 (z)? J ν+1 (z)N ν (z) =? 2 πz (E.88) H (2) ν (z)H (1) ν+1 (z)? H (1) ν (z)H (2) ν+1 (z) = 4 jπz (E.89) I ν (z)K ν+1 (z)+ I ν+1 (z)K ν (z) = 1 z (E.90) I ν (z)K prime ν (z)? I prime ν (z)K ν (z) =? 1 z (E.91) J ν (z)H (1) ν prime (z)? J prime ν (z)H (1) ν (z) = 2 j πz (E.92) J ν (z)H (2) ν prime (z)? J prime ν (z)H (2) ν (z) =? 2 j πz (E.93) H (1) ν (z)H (2) ν prime (z)? H (1) ν prime (z)H (2) ν (z) =? 4 j πz (E.94) j n (z)n n?1 (z)? j n?1 (z)n n (z) = 1 z 2 (E.95) j n+1 (z)n n?1 (z)? j n?1 (z)n n+1 (z) = 2n + 1 z 3 (E.96) j n (z)n prime n (z)? j prime n (z)n n (z) = 1 z 2 (E.97) h (1) n (z)h (2) n prime (z)? h (1) n prime (z)h (2) n (z) =? 2 j z 2 (E.98) Summationformulas ? ? ? ? ? ? ? ? ? ? ? ? ? φ ψr R ρ R, r, ρ, φ, ψ as shown. R = radicalbig r 2 +ρ 2 ? 2rρ cos φ. e jνψ Z ν (zR) = ∞ summationdisplay k=?∞ J k (zρ)Z ν+k (zr)e jkφ ,ρ<r, 0 <ψ< π 2 (E.99) e jnψ J n (zR) = ∞ summationdisplay k=?∞ J k (zρ)J n+k (zr)e jkφ (E.100) e jzρ cos φ = ∞ summationdisplay k=0 j k (2k + 1)j k (zρ)P k (cos φ) (E.101) For ρ<r and 0 <ψ<π/2, e jzR R = jπ 2 √ rρ ∞ summationdisplay k=0 (2k + 1)J k+ 1 2 (zρ)H (1) k+ 1 2 (zr)P k (cos φ) (E.102) e ?jzR R =? jπ 2 √ rρ ∞ summationdisplay k=0 (2k + 1)J k+ 1 2 (zρ)H (2) k+ 1 2 (zr)P k (cos φ) (E.103) Integrals integraldisplay x ν+1 J ν (x) dx = x ν+1 J ν+1 (x)+ C (E.104) integraldisplay Z ν (ax)Z ν (bx)xdx= x [bZ ν (ax)Z ν?1 (bx)? aZ ν?1 (ax)Z ν (bx)] a 2 ? b 2 + C, a negationslash= b (E.105) integraldisplay xZ 2 ν (ax) dx = x 2 2 bracketleftbig Z 2 ν (ax)? Z ν?1 (ax)Z ν+1 (ax) bracketrightbig + C (E.106) integraldisplay ∞ 0 J ν (ax) dx = 1 a ,ν>?1, a > 0 (E.107) Fourier–Besselexpansionofafunction f (ρ) = ∞ summationdisplay m=1 a m J ν parenleftBig p νm ρ a parenrightBig , 0 ≤ ρ ≤ a,ν>?1 (E.108) a m = 2 a 2 J 2 ν+1 (p νm ) integraldisplay a 0 f (ρ)J ν parenleftBig p νm ρ a parenrightBig ρ dρ (E.109) f (ρ) = ∞ summationdisplay m=1 b m J ν parenleftBig p prime νm ρ a parenrightBig , 0 ≤ ρ ≤ a,ν>?1 (E.110) b m = 2 a 2 parenleftBig 1 ? ν 2 p prime 2 νm J 2 ν (p prime νm ) parenrightBig integraldisplay a 0 f (ρ)J ν parenleftbigg p prime νm a ρ parenrightbigg ρ dρ (E.111) SeriesofBesselfunctions e jzcos φ = ∞ summationdisplay k=?∞ j k J k (z)e jkφ (E.112) e jzcos φ = J 0 (z)+ 2 ∞ summationdisplay k=1 j k J k (z) cos φ (E.113) sin z = 2 ∞ summationdisplay k=0 (?1) k J 2k+1 (z) (E.114) cos z = J 0 (z)+ 2 ∞ summationdisplay k=1 (?1) k J 2k (z) (E.115) E.2 Legendre functions Notation x, y,θ = real numbers; l, m, n = integers; P m n (cos θ) = associated Legendre function of the ?rst kind Q m n (cos θ) = associated Legendre function of the second kind P n (cos θ) = P 0 n (cos θ)= Legendre polynomial Q n (cos θ) = Q 0 n (cos θ)= Legendre function of the second kind Di?erentialequation x = cos θ. (1 ? x 2 ) d 2 R m n (x) dx 2 ? 2x dR m n (x) dx + bracketleftbigg n(n + 1)? m 2 1 ? x 2 bracketrightbigg R m n (x) = 0, ?1 ≤ x ≤ 1 (E.116) R m n (x) = braceleftbigg P m n (x) Q m n (x) (E.117) Orthogonalityrelationships integraldisplay 1 ?1 P m l (x)P m n (x) dx = δ ln 2 2n + 1 (n + m)! (n ? m)! (E.118) integraldisplay π 0 P m l (cos θ)P m n (cos θ)sin θ dθ = δ ln 2 2n + 1 (n + m)! (n ? m)! (E.119) integraldisplay 1 ?1 P m n (x)P k n (x) 1 ? x 2 dx = δ mk 1 m (n + m)! (n ? m)! (E.120) integraldisplay π 0 P m n (cos θ)P k n (cos θ) sin θ dθ = δ mk 1 m (n + m)! (n ? m)! (E.121) integraldisplay 1 ?1 P l (x)P n (x) dx = δ ln 2 2n + 1 (E.122) integraldisplay π 0 P l (cos θ)P n (cos θ)sin θ dθ = δ ln 2 2n + 1 (E.123) Speci?cexamples P 0 (x) = 1 (E.124) P 1 (x) = x = cos(θ) (E.125) P 2 (x) = 1 2 (3x 2 ? 1) = 1 4 (3 cos 2θ + 1) (E.126) P 3 (x) = 1 2 (5x 3 ? 3x) = 1 8 (5 cos 3θ + 3 cos θ) (E.127) P 4 (x) = 1 8 (35x 4 ? 30x 2 + 3) = 1 64 (35 cos 4θ + 20 cos 2θ + 9) (E.128) P 5 (x) = 1 8 (63x 5 ? 70x 3 + 15x) = 1 128 (63 cos 5θ + 35 cos 3θ + 30 cos θ) (E.129) Q 0 (x) = 1 2 ln parenleftbigg 1 + x 1 ? x parenrightbigg = ln parenleftbigg cot θ 2 parenrightbigg (E.130) Q 1 (x) = x 2 ln parenleftbigg 1 + x 1 ? x parenrightbigg ? 1 = cos θ ln parenleftbigg cot θ 2 parenrightbigg ? 1 (E.131) Q 2 (x) = 1 4 (3x 2 ? 1) ln parenleftbigg 1 + x 1 ? x parenrightbigg ? 3 2 x (E.132) Q 3 (x) = 1 4 (5x 3 ? 3x) ln parenleftbigg 1 + x 1 ? x parenrightbigg ? 5 2 x 2 + 2 3 (E.133) Q 4 (x) = 1 16 (35x 4 ? 30x 2 + 3) ln parenleftbigg 1 + x 1 ? x parenrightbigg ? 35 8 x 3 + 55 24 x (E.134) P 1 1 (x) =?(1 ? x 2 ) 1/2 =?sin θ (E.135) P 1 2 (x) =?3x(1 ? x 2 ) 1/2 =?3 cos θ sin θ (E.136) P 2 2 (x) = 3(1 ? x 2 ) = 3 sin 2 θ (E.137) P 1 3 (x) =? 3 2 (5x 2 ? 1)(1 ? x 2 ) 1/2 =? 3 2 (5 cos 2 θ ? 1) sin θ (E.138) P 2 3 (x) = 15x(1 ? x 2 ) = 15 cos θ sin 2 θ (E.139) P 3 3 (x) =?15(1 ? x 2 ) 3/2 =?15 sin 3 θ (E.140) P 1 4 (x) =? 5 2 (7x 3 ? 3x)(1 ? x 2 ) 1/2 =? 5 2 (7 cos 3 θ ? 3 cos θ)sin θ (E.141) P 2 4 (x) = 15 2 (7x 2 ? 1)(1 ? x 2 ) = 15 2 (7 cos 2 θ ? 1) sin 2 θ (E.142) P 3 4 (x) =?105x(1 ? x 2 ) 3/2 =?105 cos θ sin 3 θ (E.143) P 4 4 (x) = 105(1 ? x 2 ) 2 = 105 sin 4 θ (E.144) Functionalrelationships P m n (x) = braceleftBigg 0, m > n, (?1) m (1?x 2 ) m/2 2 n n! d n+m (x 2 ?1) n dx n+m , m ≤ n. (E.145) P n (x) = 1 2 n n! d n (x 2 ? 1) n dx n (E.146) R m n (x) = (?1) m (1 ? x 2 ) m/2 d m R n (x) dx m (E.147) P ?m n (x) = (?1) m (n ? m)! (n + m)! P m n (x) (E.148) P n (?x) = (?1) n P n (x) (E.149) Q n (?x) = (?1) n+1 Q n (x) (E.150) P m n (?x) = (?1) n+m P m n (x) (E.151) Q m n (?x) = (?1) n+m+1 Q m n (x) (E.152) P m n (1) = braceleftBigg 1, m = 0, 0, m > 0. (E.153) |P n (x)|≤P n (1) = 1 (E.154) P n (0) = Gamma1 parenleftbig n 2 + 1 2 parenrightbig √ πGamma1 parenleftbig n 2 + 1 parenrightbig cos nπ 2 (E.155) P ?m n (x) = (?1) m (n ? m)! (n + m)! P m n (x) (E.156) Powerseries P n (x) = n summationdisplay k=0 (?1) k (n + k)! (n ? k)!(k!) 2 2 k+1 bracketleftbig (1 ? x) k +(?1) n (1 + x) k bracketrightbig (E.157) Recursionrelationships (n + 1 ? m)R m n+1 (x)+(n + m)R m n?1 (x) = (2n + 1)xR m n (x) (E.158) (1 ? x 2 )R m n prime (x) = (n + 1)xR m n (x)?(n ? m + 1)R m n+1 (x) (E.159) (2n + 1)xR n (x) = (n + 1)R n+1 (x)+ nR n?1 (x) (E.160) (x 2 ? 1)R prime n (x) = (n + 1)[R n+1 (x)? xR n (x)] (E.161) R prime n+1 (x)? R prime n?1 (x) = (2n + 1)R n (x) (E.162) Integralrepresentations P n (cos θ)= √ 2 π integraldisplay π 0 sin parenleftbig n + 1 2 parenrightbig u √ cos θ ? cos u du (E.163) P n (x) = 1 π integraldisplay π 0 bracketleftbig x +(x 2 ? 1) 1/2 cos θ bracketrightbig n dθ (E.164) Additionformula P n (cos γ)= P n (cos θ)P n (cos θ prime )+ + 2 n summationdisplay m=1 (n ? m)! (n + m)! P m n (cos θ)P m n (cos θ prime ) cos m(φ ?φ prime ), (E.165) cos γ = cos θ cos θ prime + sin θ sin θ prime cos(φ ?φ prime ) (E.166) Summations 1 |r ? r prime | = 1 radicalbig r 2 +r prime2 ? 2rr prime cos γ = ∞ summationdisplay n=0 r n < r n+1 > P n (cos γ) (E.167) cos γ = cos θ cos θ prime + sin θ sin θ prime cos(φ ?φ prime ) (E.168) r < = min braceleftbig |r|,|r prime | bracerightbig , r > = max braceleftbig |r|,|r prime | bracerightbig (E.169) Integrals integraldisplay P n (x) dx = P n+1 (x)? P n?1 (x) 2n + 1 + C (E.170) integraldisplay 1 ?1 x m P n (x) dx = 0, m < n (E.171) integraldisplay 1 ?1 x n P n (x) dx = 2 n+1 (n!) 2 (2n + 1)! (E.172) integraldisplay 1 ?1 x 2k P 2n (x) dx = 2 2n+1 (2k)!(k + n)! (2k + 2n + 1)!(k ? n)! (E.173) integraldisplay 1 ?1 P n (x) √ 1 ? x dx = 2 √ 2 2n + 1 (E.174) integraldisplay 1 ?1 P 2n (x) √ 1 ? x 2 dx = bracketleftBigg Gamma1 parenleftbig n + 1 2 parenrightbig n! bracketrightBigg 2 (E.175) integraldisplay 1 0 P 2n+1 (x) dx = (?1) n (2n)! 2n + 2 1 (2 n n!) 2 (E.176) Fourier–Legendreseriesexpansionofafunction f (x) = ∞ summationdisplay n=0 a n P n (x), ?1 ≤ x ≤ 1 (E.177) a n = 2n + 1 2 integraldisplay 1 ?1 f (x)P n (x) dx (E.178) E.3 Spherical harmonics Notation θ,φ = real numbers; m, n = integers Y nm (θ,φ) = spherical harmonic function Di?erentialequation 1 sin θ ? ?θ parenleftbigg sin θ ?Y(θ,φ) ?θ parenrightbigg + 1 sin 2 θ ? 2 Y(θ,φ) ?φ 2 + 1 a 2 λY(θ,φ) = 0 (E.179) λ = a 2 n(n + 1) (E.180) Y nm (θ,φ) = radicalBigg 2n + 1 4π (n ? m)! (n + m)! P m n (cos θ)e jmθ (E.181) Orthogonalityrelationships integraldisplay π ?π integraldisplay π 0 Y ? n prime m prime(θ,φ)Ynm(θ,φ) sin θ dθ dφ = δn prime n δ m prime m (E.182) ∞ summationdisplay n=0 n summationdisplay m=?n Y ? nm (θ prime ,φ prime )Y nm (θ,φ) = δ(φ?φ prime )δ(cos θ ? cos θ prime ) (E.183) Speci?cexamples Y 00 (θ,φ) = radicalbigg 1 4π (E.184) Y 10 (θ,φ) = radicalbigg 3 4π cos θ (E.185) Y 11 (θ,φ) =? radicalbigg 3 8π sin θe jφ (E.186) Y 20 (θ,φ) = radicalbigg 5 4π parenleftbigg 3 2 cos 2 θ ? 1 2 parenrightbigg (E.187) Y 21 (θ,φ) =? radicalbigg 15 8π sin θ cos θe jφ (E.188) Y 22 (θ,φ) = radicalbigg 15 32π sin 2 θe 2 jφ (E.189) Y 30 (θ,φ) = radicalbigg 7 4π parenleftbigg 5 2 cos 3 θ ? 3 2 cos θ parenrightbigg (E.190) Y 31 (θ,φ) =? radicalbigg 21 64π sin θ parenleftbig 5 cos 2 θ ? 1 parenrightbig e jφ (E.191) Y 32 (θ,φ) = radicalbigg 105 32π sin 2 θ cos θe 2 jφ (E.192) Y 33 (θ,φ) =? radicalbigg 35 64π sin 3 θe 3 jφ (E.193) Functionalrelationships Y n0 (θ,φ) = radicalbigg 2n + 1 4π P n (cos θ) (E.194) Y n,?m (θ,φ) = (?1) m Y ? nm (θ,φ) (E.195) Additionformulas P n (cos γ)= 4π 2n + 1 n summationdisplay m=?n Y nm (θ,φ)Y ? nm (θ prime ,φ prime ) (E.196) P n (cos γ)= P n (cos θ)P n (cos θ prime )+ + n summationdisplay m=?n (n ? m)! (n + m)! P m n (cos θ)P m n (cos θ prime ) cos bracketleftbig m(φ ?φ prime ) bracketrightbig (E.197) cos γ = cos θ cos θ prime + sin θ sin θ prime cos(φ ?φ prime ) (E.198) Series n summationdisplay m=?n |Y nm (θ,φ)| 2 = 2n + 1 4π (E.199) 1 |r ? r prime | = 1 radicalbig r 2 +r prime2 ? 2rr prime cos γ = 4π ∞ summationdisplay n=0 n summationdisplay m=?n 1 2n + 1 r n < r n+1 > Y ? nm (θ prime ,φ prime )Y nm (θ,φ), (E.200) r < = min braceleftbig |r|,|r prime | bracerightbig , r > = max braceleftbig |r|,|r prime | bracerightbig (E.201) Seriesexpansionofafunction f (θ,φ) = ∞ summationdisplay n=0 n summationdisplay m=?n a nm Y nm (θ,φ) (E.202) a nm = integraldisplay π ?π integraldisplay π 0 f (θ,φ)Y ? nm (θ,φ) sin θ dθ dφ (E.203)