Appendix e Properties of special functions E. 1 Bessel functions Notation z= complex number; v, x= real numbers: n= integer Jv(z)=ordinary Bessel function of the first kind N,(z)= ordinary Bessel function of the second kin I,(z)=modified Bessel function of the first kind K,(z)= modified Bessel function of the second kind H(D)= Hankel function of the first kind H()= Hankel function of the second kind jn(z)=ordinary spherical Bessel function of the first kind ln(z)= ordinary spherical Bessel function of the second kind ha(z)= spherical Hankel function of the first kind h(2(z)=spherical Hankel function of the second kind f(a)=df(z)/dz= derivative with respect to argument Differential equations d-Z,(z) 1 dzv(z) x)2()=0 (E1) Zy(z) (E2) N,(z)=cos(vT)J,(z)-J-,(2) ,≠n,|arg(z)<π (E3) sin(v) Hy (2)=J,(2)+JN,(z) HQ(z)=J,(2)-jN,(z) @2001 by CRC Press LLC
Appendix E Properties of special functions E.1 Bessel functions Notation z = complex number; ν, x = real numbers; n = integer Jν (z) = ordinary Bessel function of the first kind Nν (z) = ordinary Bessel function of the second kind Iν (z) = modified Bessel function of the first kind Kν (z) = modified Bessel function of the second kind H(1) ν = Hankel function of the first kind H(2) ν = Hankel function of the second kind jn(z) = ordinary spherical Bessel function of the first kind nn(z) = ordinary spherical Bessel function of the second kind h(1) n (z) = spherical Hankel function of the first kind h(2) n (z) = spherical Hankel function of the second kind f (z) = d f (z)/dz = derivative with respect to argument Differential equations d2Zν (z) dz2 + 1 z d Zν (z) dz + 1 − ν2 z2 Zν (z) = 0 (E.1) Zν (z) = Jν (z) Nν (z) H(1) ν (z) H(2) ν (z) (E.2) Nν (z) = cos(νπ)Jν (z) − J−ν (z) sin(νπ) , ν = n, | arg(z)| < π (E.3) H(1) ν (z) = Jν (z) + j Nν (z) (E.4) H(2) ν (z) = Jν (z) − j Nν (z) (E.5)
d-Z,(x) 1 dZ,(z) (1+)2=0 (E6) l,(z) 2x)=1K(z) (E7) eKv(z) 1,(2)=ewx2J,(el12),-<ag(s→ l,()=e/3v7/2J, (e-j3x/2). T 2s arg(z) (E10) K,(e)=ejv/H(ze/ 2), -T<arg-2 (E11) K(2)=-1c-m/H(e-1/),-x<amg(x)≤r (E12) (E13) Kn(x)=jn+HOAx) (E.14) d2n()+2dn(2)+ n(n+1) zn(z)=0.n=0,±1,±2 (E15) jn (z) Zn(z) jn()=5-Jn+,(2) (E17) bg(a)=√H()=hn()+m() (E.19) 2(a)=z(o)=h()-m1() n()=(-1)+1j-an+1)(x) (E21) Orthogonality relationships C4(2mn)4(m) 24+(pm)=am2[p ], @2001 by CRC Press LLC
d2Z¯ ν (x) dz2 + 1 z d Z¯ ν (z) dz − 1 + ν2 z2 Z¯ ν = 0 (E.6) Z¯ ν (z) = Iν (z) Kν (z) (E.7) L(z) = Iν (z) e jνπ Kν (z) (E.8) Iν (z) = e− jνπ/2 Jν (ze jπ/2 ), −π −1 (E.22)
ppdp=8 Pvm J,(ax)J,(B x)xdx=-s(a-B) (E24) C0(m)(四) (E25) im(x)in(x)dx=m、2 n>0 2n+ (E26) Jm(pmn)=0 27) Jm(pm) 28 29) am=0 sIn j(z)= (E.31) (E.32) ho (z) (E.33) ho (x) cos 2 sIn 3 Functional relationships In(-z)=(-1)"ln(z) EEEE @2001 by CRC Press LLC
a 0 Jν p νm a ρ Jν p νn a ρ ρ dρ = δmn a2 2 1 − ν2 p2 νm J 2 ν (p νm), ν > −1 (E.23) ∞ 0 Jν (αx)Jν (βx)x dx = 1 α δ(α − β) (E.24) a 0 jl αlm a r jl αln a r r 2 dr = δmn a3 2 j 2 n+1(αlna) (E.25) ∞ −∞ jm(x)jn(x) dx = δmn π 2n + 1 , m, n ≥ 0 (E.26) Jm(pmn) = 0 (E.27) J m(p mn) = 0 (E.28) jm(αmn) = 0 (E.29) j m(α mn) = 0 (E.30) Specific examples j0(z) = sin z z (E.31) n0(z) = −cosz z (E.32) h(1) 0 (z) = − j z e jz (E.33) h(2) 0 (z) = j z e− jz (E.34) j1(z) = sin z z2 − cosz z (E.35) n1(z) = −cosz z2 − sin z z (E.36) j2(z) = 3 z3 − 1 z sin z − 3 z2 cosz (E.37) n2(z) = − 3 z3 + 1 z cosz − 3 z2 sin z (E.38) Functional relationships Jn(−z) = (−1) n Jn(z) (E.39) In(−z) = (−1) n In(z) (E.40) jn(−z) = (−1) n jn(z) (E.41) nn(−z) = (−1) n+1 nn(z) (E.42) J−n(z) = (−1) n Jn(z) (E.43)
1n(z)=1n(x) n(2)=Kn(z) jn(z)=(-1)"n-1(z),n>0 (E.47) Power series n(x)=)(-1) k(x/2y+2 k!(+k) (E.48) (z/2)"+ k!(n+k)! (E.49) Small argument approximations z>1 42 N(z)≈yπz ,|arg(x)<π (E60) H()=r e(--号),-m<ag()<2r H2)(z) arg(z)<丌 (E.62) 2Tz, larg(z)<I (E.63) @2001 by CRC Press LLC
N−n(z) = (−1) nNn(z) (E.44) I−n(z) = In(z) (E.45) K−n(z) = Kn(z) (E.46) j−n(z) = (−1) nnn−1(z), n > 0 (E.47) Power series Jn(z) = ∞ k=0 (−1) k (z/2)n+2k k!(n + k)! (E.48) In(z) = ∞ k=0 (z/2)n+2k k!(n + k)! (E.49) Small argument approximations |z| 1. Jn(z) ≈ 1 n! z 2 n (E.50) Jν (z) ≈ 1 (ν + 1) z 2 ν (E.51) N0(z) ≈ 2 π (ln z + 0.5772157 − ln 2) (E.52) Nn(z) ≈ −(n − 1)! π 2 z n , n > 0 (E.53) Nν (z) ≈ −(ν) π 2 z ν , ν> 0 (E.54) In(z) ≈ 1 n! z 2 n (E.55) Iν (z) ≈ 1 (ν + 1) z 2 ν (E.56) jn(z) ≈ 2nn! (2n + 1)! zn (E.57) nn(z) ≈ −(2n)! 2nn! z−(n+1) (E.58) Large argument approximations |z| 1. Jν (z) ≈ 2 πz cos z − π 4 − νπ 2 , | arg(z)| < π (E.59) Nν (z) ≈ 2 πz sin z − π 4 − νπ 2 , | arg(z)| < π (E.60) H(1) ν (z) ≈ 2 πz e j(z− π 4 − νπ 2 ), −π < arg(z) < 2π (E.61) H(2) ν (z) ≈ 2 πz e− j(z− π 4 − νπ 2 ), −2π < arg(z)<π (E.62) Iν (z) ≈ 1 2πz ez , | arg(z)| < π 2 (E.63)
K()≈{ne,larg()< (E64) (2-3).larg@)l<T (E.65) nn()≈ 2 ,|arg(x川 (E66) hB(x)≈(-1)y+1y,-x<ag()<2x (E67) h(2)(x)≈fnle~人 -2π<arg(x)<丌 Recursion relationships zZp-1(2)+x2y+1()=2vZ(z) Zu-1(x)-2+1(z)=2z1(x) zz(x)+v21(z)=z2y-1(x) (E71) zZ(x)-v2(x)=-2y+1(x) (E72) zLv-l(z)-zLv+1(z)=2vLy(z) (E73) Ly-1(z)+Lp+1()=2L(z) ZL,()+VLy()=ZLy-1(z) (E75) (z)-uL2(z) zZn-1(z)+zZn+1(z)=(2n +1)zn(z) nzn-1(z)-(n+1)xn+1(x)=(2n+1)zn(x) (E78) zzn(z)+(n+1)zn(z)=zZn-1(z) (E79) zzn(z)+nzn(z)=ZZn+1(z) (E80) Integral representations J,(z)= (E81) Jn(z)= (ne -z sin e)de .82 h()s1 e/cos cos(n0)de In(z)= e-cose cos(ne)de (E84) Kn(2= e-icosho cosh(nn)dt, arg(2)T jn(2)2n+In!/ cos(z cos 0)sin2n+1 de (z) eJz cose Pn(cos 8)sin 6 de @2001 by CRC Press LLC
Kν (z) ≈ π 2z e−z , | arg(z)| < 3π 2 (E.64) jn(z) ≈ 1 z sin z − nπ 2 , | arg(z)| < π (E.65) nn(z) ≈ −1 z cos z − nπ 2 , | 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) Recursion relationships zZν−1(z) + zZν+1(z) = 2νZν (z) (E.69) Zν−1(z) − Zν+1(z) = 2Z ν (z) (E.70) zZ ν (z) + νZν (z) = zZν−1(z) (E.71) zZ ν (z) − νZν (z) = −zZν+1(z) (E.72) zLν−1(z) − zLν+1(z) = 2νLν (z) (E.73) Lν−1(z) + Lν+1(z) = 2L ν (z) (E.74) zL ν (z) + νLν (z) = zLν−1(z) (E.75) zL ν (z) − νLν (z) = zLν+1(z) (E.76) zzn−1(z) + zzn+1(z) = (2n + 1)zn(z) (E.77) nzn−1(z) − (n + 1)zn+1(z) = (2n + 1)z n(z) (E.78) zz n(z) + (n + 1)zn(z) = zzn−1(z) (E.79) −zz n(z) + nzn(z) = zzn+1(z) (E.80) Integral representations Jn(z) = 1 2π π −π e− jnθ+ jz sin θ dθ (E.81) Jn(z) = 1 π π 0 cos(nθ − z sin θ) dθ (E.82) Jn(z) = 1 2π j −n π −π e jz cos θ cos(nθ) dθ (E.83) In(z) = 1 π π 0 ez cos θ cos(nθ) dθ (E.84) Kn(z) = ∞ 0 e−z cosh(t) cosh(nt) dt, | arg(z)| < π 2 (E.85) jn(z) = zn 2n+1n! π 0 cos(z cos θ)sin2n+1 θ dθ (E.86) jn(z) = (− j)n 2 π 0 e jz cos θ Pn(cos θ)sin θ dθ (E.87)
J,(z)Nu+1(z)-Ju+1(z)N,(z) H2(x)H(+1()-1、(B+1(x)=j (E89) l(z)kp+1(x)+l+1(z)K1(x) (E90) I,(zK (z)-I()K,(z) (E91) J(z)H((x)-(x)h()s2 (E.92) J,(z)H(2(2)-J'()H 2(2) HO()H(2)(2)-H(()H((z) (E.95) 2n+1 jin+1()n-1(x)-i-1(z)n+1(z) n()n(x)-n()n(x)= (E97) R,r,p,中 y as shown Summation formulas R=√r2+p2-2 rp cosφ Jk(zp)Zu+k (er)e k=-0 nψJn(R)= Jk(zp)Jn+k(zr)elko 产(2k+1)(p)P(cos中) (E101) k=0 Forp<rand0<ψ<x/2 (2k+1)J+4()H(H(z)P(cos) ∑2k+1)4+()H+(xr)P(os) (E.103) @2001 by CRC Press LLC
Wronskians and cross products Jν (z)Nν+1(z) − Jν+1(z)Nν (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) Iν (z)Kν+1(z) + Iν+1(z)Kν (z) = 1 z (E.90) Iν (z)K ν (z) − I ν (z)Kν (z) = −1 z (E.91) Jν (z)H(1) ν (z) − J ν (z)H(1) ν (z) = 2 j πz (E.92) Jν (z)H(2) ν (z) − J ν (z)H(2) ν (z) = − 2 j πz (E.93) H(1) ν (z)H(2) ν (z) − H(1) ν (z)H(2) ν (z) = − 4 j πz (E.94) jn(z)nn−1(z) − jn−1(z)nn(z) = 1 z2 (E.95) jn+1(z)nn−1(z) − jn−1(z)nn+1(z) = 2n + 1 z3 (E.96) jn(z)n n(z) − j n(z)nn(z) = 1 z2 (E.97) h(1) n (z)h(2) n (z) − h(1) n (z)h(2) n (z) = −2 j z2 (E.98) Summation formulas ✁ ✁ ✁ ✁ ✦✦✦✦✦✦✦✦✦✦ φ r ψ R ρ R, r, ρ, φ, ψ as shown. R = r 2 + ρ2 − 2rρ cos φ. e jνψ Zν (z R) = ∞ k=−∞ Jk (zρ)Zν+k (zr)e jkφ, ρ< r, 0 <ψ< π 2 (E.99) e jnψ Jn(z R) = ∞ k=−∞ Jk (zρ)Jn+k (zr)e jkφ (E.100) e jzρ cos φ = ∞ k=0 j k (2k + 1)jk (zρ)Pk (cos φ) (E.101) For ρ < r and 0 < ψ < π/2, e jzR R = jπ 2 √rρ ∞ k=0 (2k + 1)Jk+ 1 2 (zρ)H(1) k+ 1 2 (zr)Pk (cos φ) (E.102) e− jzR R = − jπ 2 √rρ ∞ k=0 (2k + 1)Jk+ 1 2 (zρ)H(2) k+ 1 2 (zr)Pk (cos φ) (E.103)
v+1 Zy(ax)z,(bx)x dx bZv(ax)Zy-I(bx)-aZu-l(ax)Z,(bx) +C,a≠b(E.105) [2(ax)-2-1ax)Z+1(ax)]+C Jv(ax)d Fourier-Bessel expansion of a function f(p) 0≤p≤ 108 22+1(P f(p)Jv(Pom )pdp (E109) 0≤p≤ f(p)Jv (E111) Series of bessel functions 产人 (E112) ∑ sinz=2 1)J2k+1(z) cosz=M(z)+2∑(-1)h2() E 2 Legendre functions (cos 0)= associated Legendre function of the first kind @2001 by CRC Press LLC
Integrals x ν+1 Jν (x) dx = x ν+1 Jν+1(x) + C (E.104) Zν (ax)Zν (bx)x dx = x [bZν (ax)Zν−1(bx) − aZν−1(ax)Zν (bx)] a2 − b2 + C, a = b (E.105) x Z2 ν (ax) dx = x 2 2 Z2 ν (ax) − Zν−1(ax)Zν+1(ax) + C (E.106) ∞ 0 Jν (ax) dx = 1 a , ν> −1, a > 0 (E.107) Fourier–Bessel expansion of a function f (ρ) = ∞ m=1 am Jν pνm ρ a , 0 ≤ ρ ≤ a, ν> −1 (E.108) am = 2 a2 J 2 ν+1(pνm) a 0 f (ρ)Jν pνm ρ a ρ dρ (E.109) f (ρ) = ∞ m=1 bm Jν p νm ρ a , 0 ≤ ρ ≤ a, ν> −1 (E.110) bm = 2 a2 1 − ν2 p2 νm J 2 ν (p νm) a 0 f (ρ)Jν p νm a ρ ρ dρ (E.111) Series of Bessel functions e jz cos φ = ∞ k=−∞ j k Jk (z)e jkφ (E.112) e jz cos φ = J0(z) + 2 ∞ k=1 j k Jk (z) cos φ (E.113) sin z = 2 ∞ k=0 (−1) k J2k+1(z) (E.114) cosz = J0(z) + 2 ∞ k=1 (−1) k J2k (z) (E.115) E.2 Legendre functions Notation x, y, θ = real numbers; l, m, n = integers; Pm n (cos θ) = associated Legendre function of the first kind
On(cos 0)= associated Legendre function of the second kind Pn(cos 0)= Pn(cos 0)= Legendre polynomial On(cos 0)= qm(cos 0)= Legendre function of the second kind Differential equation x=cos 6 dx2 +|n(n+1) 1]<()=0.-1x110 Rm()=gm(x) (E117) Orthogonality relationships 2(n+m)! Pm(x)P(x)dx=Sin 2n+I(n-m)! (E118) Pi(cos 0)Pn (cos 8)sin 8 de 8I 2(n+m)! 2n+1(n-m)! (E119) Pm(x)Ps(④dx=8n1(n+m (n-m)! (E120) P(cos 8)P (cos 0) 1(n +m) (E121) (E122) P(cos 0)Pn(cos 0)sin e de =2+1 (E123) Specific examples P(x)=1 (E124) PI(x)=x= cos(0) (E125) P2(x 3x2-1)=-(3cos26+1) (E126) P()=1(5x3-3x)=15c0s3+3c0s6 P2(x)=(35x4-30x2+3)=(35cos46+20cs26+9) (E128) P()=3(6x32-70x3+15x)=12(63c0s56+35c36+30c0)(E.2) Oo(x)=In =In/co, 8 (E.130) -1=cos 0 In=-I (E131) @2001 by CRC Press LLC
Qm n (cos θ) = associated Legendre function of the second kind Pn(cos θ) = P0 n (cos θ) = Legendre polynomial Qn(cos θ) = Q0 n(cos θ) = Legendre function of the second kind Differential equation x = cos θ. (1 − x 2 ) d2Rm n (x) dx 2 − 2x d Rm n (x) dx + n(n + 1) − m2 1 − x 2 Rm n (x) = 0, −1 ≤ x ≤ 1 (E.116) Rm n (x) = Pm n (x) Qm n (x) (E.117) Orthogonality relationships 1 −1 Pm l (x)Pm n (x) dx = δln 2 2n + 1 (n + m)! (n − m)! (E.118) π 0 Pm l (cos θ)Pm n (cos θ)sin θ dθ = δln 2 2n + 1 (n + m)! (n − m)! (E.119) 1 −1 Pm n (x)Pk n (x) 1 − x 2 dx = δmk 1 m (n + m)! (n − m)! (E.120) π 0 Pm n (cos θ)Pk n (cos θ) sin θ dθ = δmk 1 m (n + m)! (n − m)! (E.121) 1 −1 Pl(x)Pn(x) dx = δln 2 2n + 1 (E.122) π 0 Pl(cos θ)Pn(cos θ)sin θ dθ = δln 2 2n + 1 (E.123) Specific examples P0(x) = 1 (E.124) P1(x) = x = cos(θ) (E.125) P2(x) = 1 2 (3x 2 − 1) = 1 4 (3 cos 2θ + 1) (E.126) P3(x) = 1 2 (5x 3 − 3x) = 1 8 (5 cos 3θ + 3 cos θ) (E.127) P4(x) = 1 8 (35x 4 − 30x 2 + 3) = 1 64(35 cos 4θ + 20 cos 2θ + 9) (E.128) P5(x) = 1 8 (63x 5 − 70x 3 + 15x) = 1 128(63 cos 5θ + 35 cos 3θ + 30 cos θ) (E.129) Q0(x) = 1 2 ln1 + x 1 − x = ln cot θ 2 (E.130) Q1(x) = x 2 ln1 + x 1 − x − 1 = cos θ ln cot θ 2 − 1 (E.131)
Q2(x) (3x2-1)h/+x)3 Q3(x)=(5x3-3x)ln (E133) Q4(x)=1(35x4-30-22、,(1+ Pl(x)=-(1-x2)2=-sin6 (E.135) P(x)=-3x(1-x2)2=-3cos6sin (E.136) P2(x)=3(1-x2)=3sin2 (E137) P(x)=-2(5x2-1)-x35o6-1)in E.138) P2(x)=15x(1-x2)=15cos6sin26 (E.139) P3(x)=-15(1-x2)312=-15sin36 (E140) P(x)=-7(7x3-3x)(1-x2)2=-5(7cos6-3cos)sinb(E141) P2(x)=(7x2-1)(1-x2)==(7 P2(x)=-105x(1-x2)312=-105cos6sin36 (E143) P4(x)=105(1-x2)2=105sin16 (E144) Functional relationships m >n. -1a, (E145) Pn(x)=2"n! dxn Rm(x)=(-1)=(1-22d8n x)= (n +mi pn(r) (E148) Pn(-x)=(-1)Pn(x) Qn(-x)=(-1)y+gn(x) (E.150) Pn(-x)=(-1)”+mPm(x) (E151) Qm(-x)=(-1)+m+gm(x) (E152) (1)= 0.m>0 Pn(x)≤Pn(1) @2001 by CRC Press LLC
Q2(x) = 1 4 (3x 2 − 1)ln1 + x 1 − x − 3 2 x (E.132) Q3(x) = 1 4 (5x 3 − 3x)ln1 + x 1 − x − 5 2 x 2 + 2 3 (E.133) Q4(x) = 1 16(35x 4 − 30x 2 + 3)ln1 + x 1 − x − 35 8 x 3 + 55 24 x (E.134) P1 1 (x) = −(1 − x 2 ) 1/2 = − sin θ (E.135) P1 2 (x) = −3x(1 − x 2 ) 1/2 = −3 cos θ sin θ (E.136) P2 2 (x) = 3(1 − x 2 ) = 3 sin2 θ (E.137) P1 3 (x) = −3 2 (5x 2 − 1)(1 − x 2 ) 1/2 = −3 2 (5 cos2 θ − 1)sin θ (E.138) P2 3 (x) = 15x(1 − x 2 ) = 15 cos θ sin2 θ (E.139) P3 3 (x) = −15(1 − x 2 ) 3/2 = −15 sin3 θ (E.140) P1 4 (x) = −5 2 (7x 3 − 3x)(1 − x 2 ) 1/2 = −5 2 (7 cos3 θ − 3 cos θ)sin θ (E.141) P2 4 (x) = 15 2 (7x 2 − 1)(1 − x 2 ) = 15 2 (7 cos2 θ − 1)sin2 θ (E.142) P3 4 (x) = −105x(1 − x 2 ) 3/2 = −105 cos θ sin3 θ (E.143) P4 4 (x) = 105(1 − x 2 ) 2 = 105 sin4 θ (E.144) Functional relationships Pm n (x) = 0, m > n, (−1)m (1−x2)m/2 2n n! dn+m (x2−1)n dxn+m , m ≤ n. (E.145) Pn(x) = 1 2nn! dn(x 2 − 1)n dx n (E.146) Rm n (x) = (−1) m(1 − x 2 ) m/2 dm Rn(x) dxm (E.147) P−m n (x) = (−1) m (n − m)! (n + m)! Pm n (x) (E.148) Pn(−x) = (−1) n Pn(x) (E.149) Qn(−x) = (−1) n+1Qn(x) (E.150) Pm n (−x) = (−1) n+m Pm n (x) (E.151) Qm n (−x) = (−1) n+m+1Qm n (x) (E.152) Pm n (1) = 1, m = 0, 0, m > 0. (E.153) |Pn(x)| ≤ Pn(1) = 1 (E.154)
r(号+ Pn(0) (E155) Pn"(x)=(-1) (-m) P(x) P(x)= ∑m-6)(k92+(-x)+(-1+门]E157) (n+1-m)Rn+1(x)+(n+m)Rn1(x)=(2n+1)xBm(x) (E158) (1-x2)Ram(x)=(n+1)xRm(x)-(n-m+1)Rm+1(x) (E159) (2n+1)xRn(x)=(+1)Rn+1(x)+nRn-1(x) (E.160) (x2-1)R(x)=(n+1)Rn+1(x)-xRn(x) Rn+1(x)-Rn-1(x)=(2n+1)Rn(x) (E162) Integral representations √cosb-cos Pn(x) Addition formula Pn(cos y)= Pn(cos 0)Pn(cos 8)+ +2 (n-m)! (φ一中) cosy= cos e cos6′+ sin 0 sin6’cos(φ-φ) (E.166) (E.167) rrcosy n=0/> cosy= cos e cos6+ sin e sin 0 cos(φ-φ) (E.168) GIrl, Irll GIrl, Irll @2001 by CRC Press LLC
Pn(0) = n 2 + 1 2 √π n 2 + 1 cos nπ 2 (E.155) P−m n (x) = (−1) m (n − m)! (n + m)! Pm n (x) (E.156) Power series Pn(x) = n k=0 (−1)k (n + k)! (n − k)!(k!)22k+1 (1 − x) k + (−1) n(1 + x) k (E.157) Recursion relationships (n + 1 − m)Rm n+1(x) + (n + m)Rm n−1(x) = (2n + 1)x Rm n (x) (E.158) (1 − x 2 )Rm n (x) = (n + 1)x Rm n (x) − (n − m + 1)Rm n+1(x) (E.159) (2n + 1)x Rn(x) = (n + 1)Rn+1(x) + n Rn−1(x) (E.160) (x 2 − 1)R n(x) = (n + 1)[Rn+1(x) − x Rn(x)] (E.161) R n+1(x) − R n−1(x) = (2n + 1)Rn(x) (E.162) Integral representations Pn(cos θ) = √2 π π 0 sin n + 1 2 u √cos θ − cos u du (E.163) Pn(x) = 1 π π 0 x + (x 2 − 1) 1/2 cos θ n dθ (E.164) Addition formula Pn(cos γ) = Pn(cos θ)Pn(cos θ ) + + 2 n m=1 (n − m)! (n + m)! Pm n (cos θ)Pm n (cos θ ) cos m(φ − φ ), (E.165) cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.166) Summations 1 |r − r | = 1 r 2 + r2 − 2rr cos γ = ∞ n=0 r n Pn(cos γ) (E.167) cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.168) r = max |r|, |r | (E.169)
