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10 Bessel FunctionsModified Bessel Functions

§10.42 Zeros

Properties of the zeros of Iν(z) and Kν(z) may be deduced from those of Jν(z) and Hν(1)(z), respectively, by application of the transformations (10.27.6) and (10.27.8).

For example, if ν is real, then the zeros of Iν(z) are all complex unless -2<ν<-(2-1) for some positive integer , in which event Iν(z) has two real zeros.

The distribution of the zeros of Kn(nz) in the sector -32πphz12π in the cases n=1,5,10 is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle -12π so that in each case the cut lies along the positive imaginary axis. The zeros in the sector -12πphz32π are their conjugates.

Kn(z) has no zeros in the sector |phz|12π; this result remains true when n is replaced by any real number ν. For the number of zeros of Kν(z) in the sector |phz|π, when ν is real, see Watson (1944, pp. 511–513).

For z-zeros of Kν(z), with complex ν, see Ferreira and Sesma (2008).

See also Kerimov and Skorokhodov (1984b, a).