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 LLCKν (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)