# §13.9 Zeros

## §13.9(i) Zeros of

If and , then has infinitely many -zeros in . When the number of real zeros is finite. Let be the number of positive zeros. Then

13.9.1, ,
13.9.2, ,
13.9.3, ,
13.9.4, .
13.9.5, , ,

The number of negative real zeros is given by

13.9.7

When and let , , be the positive zeros of arranged in increasing order of magnitude, and let be the th positive zero of the Bessel function 10.21(i)). Then

as with fixed.

Inequalities for are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of are given in Ahmed and Muldoon (1980).

For fixed the large -zeros of satisfy

where is a large positive integer, and the logarithm takes its principal value (§4.2(i)).

Let denote the closure of the domain that is bounded by the parabola and contains the origin. Then has no zeros in the regions , if ; , if ; , where , if and . The same results apply for the th partial sums of the Maclaurin series (13.2.2) of .

More information on the location of real zeros can be found in Zarzo et al. (1995) and Segura (2008).

For fixed and in the large -zeros of are given by

where is a large positive integer.

For fixed and in the function has only a finite number of -zeros.

## §13.9(ii) Zeros of

For fixed and in , has a finite number of -zeros in the sector . Let be the total number of zeros in the sector , be the corresponding number of positive zeros, and , , and be nonintegers. For the case

13.9.11, ,
13.9.12, ,
13.9.13,

and

13.9.14,
13.9.15.

For the case we can use and .

In Wimp (1965) it is shown that if and , then has no zeros in the sector .

Inequalities for the zeros of are given in Gatteschi (1990). See also Segura (2008).

For fixed and in the large -zeros of are given by

where is a large positive integer.

For fixed and in , has two infinite strings of -zeros that are asymptotic to the imaginary axis as .