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 < arg(z) ≤ π 2 (E.9) Iν (z) = e j3νπ/2 Jν (ze− j3π/2 ), π 2 < arg(z) ≤ π (E.10) Kν (z) = jπ 2 e jνπ/2H(1) ν (ze jπ/2 ), −π < arg(z) ≤ π 2 (E.11) Kν (z) = − jπ 2 e− jνπ/2H(2) ν (ze− jπ/2 ), −π 2 < arg(z) ≤ π (E.12) In(x) = j −n Jn(j x) (E.13) Kn(x) = π 2 j n+1H(1) n (j x) (E.14) d2zn(z) dz2 + 2 z dzn(z) dz + 1 − n(n + 1) z2 zn(z) = 0, n = 0, ±1, ±2,... (E.15) zn(z) = jn(z) nn(z) h(1) n (z) h(2) n (z) (E.16) jn(z) = π 2z Jn+ 1 2 (z) (E.17) nn(z) = π 2z Nn+ 1 2 (z) (E.18) h(1) n (z) = π 2z H(1) n+ 1 2 (z) = jn(z) + jnn(z) (E.19) h(2) n (z) = π 2z H(2) n+ 1 2 (z) = jn(z) − jnn(z) (E.20) nn(z) = (−1) n+1 j−(n+1)(z) (E.21) Orthogonality relationships a 0 Jν pνm a ρ Jν pνn a ρ ρ dρ = δmn a2 2 J 2 ν+1(pνn) = δmn a2 2 J ν (pνn) 2 , ν> −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 Jn(z) J,(z) r(v+1) (E51) No(z)≈-(lnx+0.5772157-ln2) (E52) Nn(z) (E53) T(v) In(z) 1 l(x)≈ r(v+1)(2 (E57) (2n+1) (2n)! (n+1) Large argument approximations zI >>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)