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10 Bessel FunctionsSpherical Bessel Functions

§10.58 Zeros

For n0 the mth positive zeros of jn(x), jn(x), yn(x), and yn(x) are denoted by an,m, an,m, bn,m, and bn,m, respectively, except that for n=0 we count x=0 as the first zero of j0(x).

With the notation of §10.21(i),

10.58.1 an,m =jn+12,m,
bn,m =yn+12,m,
10.58.2 jn(an,m) =π2jn+12,mJn+12(jn+12,m),
yn(bn,m) =π2yn+12,mYn+12(yn+12,m).

Hence properties of an,m and bn,m are derivable straightforwardly from results given in §§10.21(i)10.21(iii), 10.21(vi)10.21(viii), and 10.21(x). However, there are no simple relations that connect the zeros of the derivatives. For some properties of an,m and bn,m, including asymptotic expansions, see Olver (1960, pp. xix–xxi).

See also Davies (1973), de Bruin et al. (1981a, b), and Gottlieb (1985).