# §8.13 Zeros

## §8.13(i) -Zeros of

The function has no real zeros for . For and , there exist:

1. (a)

one negative zero and no positive zeros when ;

2. (b)

one negative zero and one positive zero when .

The negative zero decreases monotonically in the interval , and satisfies

When the behavior of the -zeros as functions of can be seen by taking the slice of the surface depicted in Figure 8.3.6. Note that from (8.4.12) , .

For asymptotic approximations for and as see Tricomi (1950b), with corrections by Kölbig (1972b).

## §8.13(ii) -Zeros of and

For information on the distribution and computation of zeros of and in the complex -plane for large values of the positive real parameter see Temme (1995a).

## §8.13(iii) -Zeros of

For fixed and , has:

1. (a)

two zeros in each of the intervals when ;

2. (b)

two zeros in each of the intervals when ;

3. (c)

zeros at when .

As increases the positive zeros coalesce to form a double zero at (). The values of the first six double zeros are given to 5D in Table 8.13.1. For values up to see Kölbig (1972b). Approximations to , for large can be found in Kölbig (1970). When a pair of conjugate trajectories emanate from the point in the complex -plane. See Kölbig (1970, 1972b) for further information.

Table 8.13.1: Double zeros of .
1 −1.64425 0.30809
2 −3.63887 0.77997
3 −5.63573 1.28634
4 −7.63372 1.80754
5 −9.63230 2.33692
6 −11.63126 2.87150