Digital Library of Mathematical Functions
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10 Bessel FunctionsKelvin Functions

§10.70 Zeros

Asymptotic approximations for large zeros are as follows. Let μ=4ν2 and f(t) denote the formal series

10.70.1 μ-116t+μ-132t2+(μ-1)(5μ+19)1536t3+3(μ-1)2512t4+.

If m is a large positive integer, then

10.70.2 zeros of berνx 2(t-f(t)),
t=(m-12ν-38)π,
zeros of beiνx 2(t-f(t)),
t=(m-12ν+18)π,
zeros of kerνx 2(t+f(-t)),
t=(m-12ν-58)π,
zeros of keiνx 2(t+f(-t)),
t=(m-12ν-18)π.

In the case ν=0, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the mth zero of the function on the left-hand side. For the next six terms in the series (10.70.1) see MacLeod (2002a).