# §12.11 Zeros

## §12.11(i) Distribution of Real Zeros

If then has no real zeros. If , then has no positive real zeros. If , , then has positive real zeros. Lastly, when , (Hermite polynomial case) has zeros and they lie in the interval . For further information on these cases see Dean (1966).

If then has no positive real zeros, and if , , then has a zero at .

## §12.11(ii) Asymptotic Expansions of Large Zeros

When , has a string of complex zeros that approaches the ray as , and a conjugate string. When the zeros are asymptotically given by and , where is a large positive integer and

with

and

When these zeros are the same as the zeros of the complementary error function ; compare (12.7.5). Numerical calculations in this case show that corresponds to the th zero on the string; compare §7.13(ii).

## §12.11(iii) Asymptotic Expansions for Large Parameter

For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the th real zeros of and , counted in descending order away from the point , be denoted by and , respectively. Then

as () , fixed. Here , denoting the th negative zero of the function (see §9.9(i)). The first two coefficients are given by

12.11.5

where is the function inverse to , defined by (12.10.39) (see also (12.10.41)), and

12.11.6

Similarly, for the zeros of we have

where , denoting the th negative zero of the function and

12.11.8

For the first zero of we also have

where the numerical coefficients have been rounded off.

For further information, including associated functions, see Olver (1959).