ABSTRACT
H i ) (z) = Jm (z) + iYm (2). ( A 4 The modified Bessel functions I, (z) and Km@) are linearly-independent solutions of the linear second-order differential equation
and they are related to Jm and Ym by
244 A. BESSEL F UNCTIONS
Bessel functions satisfy certain Wronskian relations. For example
The corresponding results for small real arguments are that as X + 0 ,
I m ( z ) ( ~ / 2 ) " / m ! , (A.13) KO(%) N - In X , (A.14)
Km(.) +(m - l ) ! ( ~ / 2 ) ~ ~ , m = 1,2,3, . . . , (A.15) Jm(x ) ( ~ / 2 ) " / m ! , (A.16)
2 Y o ( z ) -- - i ~ ; l ) ( x ) -- - l n z ,
n (A.17)
For the derivatives of the Bessel functions we have that as X + 0 ,
J ; (X) N - z / 2 , (A.19) J;(x) -- 2-mxm-1/(m - l ) ! , m = 1,2,3, . . . , (A.20)
H;')'(z) 2 i /nx , (A.21) H:)/(X) N 2mim!/nzm+1, rn = 1,2,3, . . . . (A.22)
The functions Jm ( z ) , Ym ( z ) , J& ( z ) , and YA ( z ) , m = 0,1,2, . . . , all possess an infinite number of real zeros. For J,, Ym, YA, m 2 0 , and
for J A , m > 0, the nth positive zeros are labelled jmn, ymn, jk,, and yk,, respectively, but z = 0 is counted as the first zero of Jh(z). The zeros interlace:
where p = 4m2, s = (n+m/2-114)~ and t = (n+m/2-314)~. Exactly the same asymptotic expansions apply to the zeros ymn and ykn if the values of s and t are interchanged.
