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 LLCAppendix 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)