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 LLCN−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)