§7.13 Zeros

§7.13(i) Zeros of

has a simple zero at , and in the first quadrant of there is an infinite set of zeros , , arranged in order of increasing absolute value. The other zeros of are , , .

Table 7.13.1 gives 10D values of the first five and . For graphical illustration see Figure 7.3.5.

Table 7.13.1: Zeros of
1 1.45061 61632 1.88094 30002
2 2.24465 92738 2.61657 51407
3 2.83974 10469 3.17562 80996
4 3.33546 07354 3.64617 43764
5 3.76900 55670 4.06069 72339

As

7.13.1

where

§7.13(ii) Zeros of

In the sector , has an infinite set of zeros , , arranged in order of increasing absolute value. The other zeros of are . The zeros of are and .

Table 7.13.2 gives 10D values of the first five and . For graphical illustration see Figure 7.3.6.

Table 7.13.2: Zeros of
1 −1.35481 01281 1.99146 68428
2 −2.17704 49061 2.69114 90243
3 −2.78438 76132 3.23533 08684
4 −3.28741 07894 3.69730 97025
5 −3.72594 87194 4.10610 72847

§7.13(iii) Zeros of the Fresnel Integrals

At , has a simple zero and has a triple zero. In the first quadrant of has an infinite set of zeros , , arranged in order of increasing absolute value. Similarly for . Let be a zero of one of the Fresnel integrals. Then , , , , , , are also zeros of the same integral.

Tables 7.13.3 and 7.13.4 give 10D values of the first five and of and , respectively.

Table 7.13.3: Complex zeros of
1 1.74366 74862 0.30573 50636
2 2.65145 95973 0.25290 39555
3 3.32035 93363 0.22395 34581
4 3.87573 44884 0.20474 74706
5 4.36106 35170 0.19066 97324

As the and corresponding to the zeros of satisfy

7.13.5
,

with

7.13.6
Table 7.13.4: Complex zeros of .
1 2.00925 70118 0.28854 78973
2 2.83347 72325 0.24428 52408
3 3.46753 30835 0.21849 26805
4 4.00257 82433 0.20085 10251
5 4.47418 92952 0.18768 85891

As the and corresponding to the zeros of satisfy (7.13.5) with

§7.13(iv) Zeros of

In consequence of (7.5.5) and (7.5.10), zeros of are related to zeros of . Thus if is a zero of 7.13(ii)), then is a zero of .

For an asymptotic expansion of the zeros of ( ) see Tuẑilin (1971).