Digital Library of Mathematical Functions
About the Project
NIST
9 Airy and Related FunctionsAiry Functions

§9.9 Zeros

Contents

§9.9(i) Distribution and Notation

On the real line, Ai(x), Ai(x), Bi(x), Bi(x) each have an infinite number of zeros, all of which are negative. They are denoted by ak, ak, bk, bk, respectively, arranged in ascending order of absolute value for k=1,2,.

Ai(z) and Ai(z) have no other zeros. However, Bi(z) and Bi(z) each have an infinite number of complex zeros. They lie in the sectors 13π<phz<12π and -12π<phz<-13π, and are denoted by βk, βk, respectively, in the former sector, and by βk¯, βk¯, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for k=1,2,. See §9.3(ii) for visualizations.

For the distribution in of the zeros of Ai(z)-σAi(z), where σ is an arbitrary complex constant, see Muraveĭ (1976).

§9.9(ii) Relation to Modulus and Phase

§9.9(iii) Derivatives With Respect to k

If k is regarded as a continuous variable, then

9.9.5 Ai(ak) =(-1)k-1(-akk)-1/2,
Ai(ak) =(-1)k-1(akakk)-1/2.

See Olver (1954, Appendix).

§9.9(iv) Asymptotic Expansions

For large k

9.9.6 ak =-T(38π(4k-1)),
9.9.7 Ai(ak) =(-1)k-1V(38π(4k-1)),
9.9.8 ak =-U(38π(4k-3)),
9.9.9 Ai(ak) =(-1)k-1W(38π(4k-3)).
9.9.10 bk =-T(38π(4k-3)),
9.9.11 Bi(bk) =(-1)k-1V(38π(4k-3)),
9.9.12 bk =-U(38π(4k-1)),
9.9.13 Bi(bk) =(-1)kW(38π(4k-1)).
9.9.14 βk =π/3T(38π(4k-1)+34ln2),
9.9.15 Bi(βk) =(-1)k2-π/6V(38π(4k-1)+34ln2),
9.9.16 βk =π/3U(38π(4k-3)+34ln2),
9.9.17 Bi(βk) =(-1)k-12π/6W(38π(4k-3)+34ln2).

Here

9.9.18 T(t)t2/3(1+548t-2-536t-4+7712582944t-6-1080 5687569 67296t-8+16 23755 968753344 30208t-10-,)
9.9.19 U(t)t2/3(1-748t-2+35288t-4-1 812232 07360t-6+186 8337112 44160t-8-9 11458 843611911 02976t-10+,)
9.9.20 V(t)π-1/2t1/6(1+548t-2-15254608t-4+23 978756 63552t-6-7 48989 406258918 13888t-8+14419 83037 343754 28070 66624t-10-,)
9.9.21 W(t)π-1/2t-1/6(1-796t-2+16736144t-4-843 94709265 42080t-6+78 02771 354211 01921 58720t-8-20444 90510 519456 52298 15808t-10+.)

For higher terms see Fabijonas and Olver (1999).

For error bounds for the asymptotic expansions of ak, bk, ak, and bk see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999).

§9.9(v) Tables

Tables 9.9.1 and 9.9.2 give 10D values of the first ten real zeros of Ai, Ai, Bi, Bi, together with the associated values of the derivative or the function. Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of Bi and Bi in the upper half plane.

Table 9.9.1: Zeros of Ai and Ai.
k ak Ai(ak) ak Ai(ak)
1 -2.33810 74105 0.70121 08227 -1.01879 29716 0.53565 66560
2 -4.08794 94441 -0.80311 13697 -3.24819 75822 -0.41901 54780
3 -5.52055 98281 0.86520 40259 -4.82009 92112 0.38040 64686
4 -6.78670 80901 -0.91085 07370 -6.16330 73556 -0.35790 79437
5 -7.94413 35871 0.94733 57094 -7.37217 72550 0.34230 12444
6 -9.02265 08533 -0.97792 28086 -8.48848 67340 -0.33047 62291
7 -10.04017 43416 1.00437 01227 -9.53544 90524 0.32102 22882
8 -11.00852 43037 -1.02773 86888 -10.52766 03970 -0.31318 53910
9 -11.93601 55632 1.04872 06486 -11.47505 66335 0.30651 72939
10 -12.82877 67529 -1.06779 38592 -12.38478 83718 -0.30073 08293
Table 9.9.2: Real zeros of Bi and Bi.
k bk Bi(bk) bk Bi(bk)
1 -1.17371 32227 0.60195 78880 -2.29443 96826 -0.45494 43836
2 -3.27109 33028 -0.76031 01415 -4.07315 50891 0.39652 28361
3 -4.83073 78417 0.83699 10126 -5.51239 57297 -0.36796 91615
4 -6.16985 21283 -0.88947 99014 -6.78129 44460 0.34949 91168
5 -7.37676 20794 0.92998 36386 -7.94017 86892 -0.33602 62401
6 -8.49194 88465 -0.96323 44302 -9.01958 33588 0.32550 97364
7 -9.53819 43793 0.99158 63705 -10.03769 63349 -0.31693 46537
8 -10.52991 35067 -1.01638 96592 -11.00646 26677 0.30972 59408
9 -11.47695 35513 1.03849 42860 -11.93426 16450 -0.30352 76648
10 -12.38641 71386 -1.05847 18444 -12.82725 83092 0.29810 49111
Table 9.9.3: Complex zeros of Bi.
-π/3βk Bi(βk)
k modulus phase modulus phase
1 2.35387 33809 0.09533 49591 0.99310 68457 2.64060 02521
2 4.09328 73094 0.04178 55604 1.13612 83345 -0.51328 28720
3 5.52350 35011 0.02668 05442 1.22374 37881 2.62462 83591
4 6.78865 95301 0.01958 69751 1.28822 92493 -0.51871 63829
5 7.94555 90160 0.01547 08228 1.33979 47726 2.62185 44560
6 9.02375 63663 0.01278 34808 1.38303 39005 -0.52040 69437
7 10.04106 73680 0.01089 12610 1.42042 53456 2.62071 41895
8 11.00926 72579 0.00948 68445 1.45346 64633 -0.52122 87219
9 11.93664 76131 0.00840 31785 1.48313 45656 2.62009 35195
10 12.82932 39388 0.00754 16607 1.51010 46383 -0.52171 41947
Table 9.9.4: Complex zeros of Bi.
-π/3βk Bi(βk)
k modulus phase modulus phase
1 1.12139 32942 0.33072 66208 0.75004 14897 0.46597 78930
2 3.25690 82266 0.05938 99367 0.59221 66315 -2.63235 40329
3 4.82400 26102 0.03278 56423 0.53787 06321 0.51549 32992
4 6.16568 66408 0.02266 24588 0.50611 02160 -2.62362 85920
5 7.37383 79870 0.01731 96481 0.48406 00643 0.51928 28169
6 8.48973 85596 0.01401 65283 0.46734 68449 -2.62149 05716
7 9.53644 07072 0.01177 19311 0.45398 23240 0.52066 02139
8 10.52847 37502 0.01014 71783 0.44290 25018 -2.62052 78353
9 11.47574 11237 0.00891 66153 0.43347 44668 0.52137 15495
10 12.38537 59341 0.00795 22843 0.42529 25837 -2.61998 05803