# §9.9 Zeros

## §9.9(i) Distribution and Notation

On the real line, $\mathrm{Ai}\left(x\right)$, $\mathrm{Ai}'\left(x\right)$, $\mathrm{Bi}\left(x\right)$, $\mathrm{Bi}'\left(x\right)$ each have an infinite number of zeros, all of which are negative. They are denoted by $a_{k}$, $a^{\prime}_{k}$, $b_{k}$, $b^{\prime}_{k}$, respectively, arranged in ascending order of absolute value for $k=1,2,\ldots.$

$\mathrm{Ai}\left(z\right)$ and $\mathrm{Ai}'\left(z\right)$ have no other zeros. However, $\mathrm{Bi}\left(z\right)$ and $\mathrm{Bi}'\left(z\right)$ each have an infinite number of complex zeros. They lie in the sectors $\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi$ and $-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi$, and are denoted by $\beta_{k}$, $\beta^{\prime}_{k}$, respectively, in the former sector, and by $\bar{\beta_{k}}$, $\bar{\beta^{\prime}_{k}}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\ldots.$ See §9.3(ii) for visualizations.

For the distribution in $\mathbb{C}$ of the zeros of $\mathrm{Ai}'\left(z\right)-\sigma\mathrm{Ai}\left(z\right)$, where $\sigma$ is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014).

## §9.9(ii) Relation to Modulus and Phase

 9.9.1 $\displaystyle\theta\left(a_{k}\right)$ $\displaystyle=\phi\left(a^{\prime}_{k+1}\right)=k\pi,$ 9.9.2 $\displaystyle\theta\left(b_{k}\right)$ $\displaystyle=\phi\left(b^{\prime}_{k}\right)=(k-\tfrac{1}{2})\pi.$
 9.9.3 $\displaystyle\mathrm{Ai}'\left(a_{k}\right)$ $\displaystyle=\frac{(-1)^{k-1}}{\pi M\left(a_{k}\right)},$ $\displaystyle\mathrm{Bi}'\left(b_{k}\right)$ $\displaystyle=\frac{(-1)^{k-1}}{\pi M\left(b_{k}\right)},$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(\NVar{z}\right)$: Airy modulus function, $a_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Ai}$, $b_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Bi}$ and $k$: nonnegative integer Source: For the first equation combine (9.2.7) with (9.8.2) and (9.9.1), and for the second equation combine (9.2.7) with (9.8.1) and (9.9.2). Permalink: http://dlmf.nist.gov/9.9.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.9(ii), 9.9 and 9
 9.9.4 $\displaystyle\mathrm{Ai}\left(a^{\prime}_{k}\right)$ $\displaystyle=\frac{(-1)^{k-1}}{\pi N\left(a^{\prime}_{k}\right)},$ $\displaystyle\mathrm{Bi}\left(b^{\prime}_{k}\right)$ $\displaystyle=\frac{(-1)^{k}}{\pi N\left(b^{\prime}_{k}\right)}.$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $N\left(\NVar{z}\right)$: Airy modulus function, $a^{\prime}_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Ai}'$, $b^{\prime}_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Bi}'$ and $k$: nonnegative integer Source: For the first equation combine (9.2.7) with (9.8.6) and (9.9.1), and for the second equation combine (9.2.7) with (9.8.5) and (9.9.2). Permalink: http://dlmf.nist.gov/9.9.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.9(ii), 9.9 and 9

## §9.9(iii) Derivatives With Respect to $k$

If $k$ is regarded as a continuous variable, then

 9.9.5 $\displaystyle\mathrm{Ai}'\left(a_{k}\right)$ $\displaystyle=(-1)^{k-1}\left(-\frac{\mathrm{d}a_{k}}{\mathrm{d}k}\right)^{-1/% 2},$ $\displaystyle\mathrm{Ai}\left(a^{\prime}_{k}\right)$ $\displaystyle=(-1)^{k-1}\left(a^{\prime}_{k}\frac{\mathrm{d}a^{\prime}_{k}}{% \mathrm{d}k}\right)^{-1/2}.$

See Olver (1954, Appendix).

## §9.9(iv) Asymptotic Expansions

For large $k$

 9.9.6 $\displaystyle a_{k}$ $\displaystyle=-T\left(\tfrac{3}{8}\pi(4k-1)\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Ai}$, $k$: nonnegative integer and $T$: expansion Source: Olver (1954, (A 20)) Permalink: http://dlmf.nist.gov/9.9.E6 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9 9.9.7 $\displaystyle\mathrm{Ai}'\left(a_{k}\right)$ $\displaystyle=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)\right),$ 9.9.8 $\displaystyle a^{\prime}_{k}$ $\displaystyle=-U\left(\tfrac{3}{8}\pi(4k-3)\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Ai}'$, $k$: nonnegative integer and $U$: expansion Source: Olver (1954, (A 20)) Permalink: http://dlmf.nist.gov/9.9.E8 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9 9.9.9 $\displaystyle\mathrm{Ai}\left(a^{\prime}_{k}\right)$ $\displaystyle=(-1)^{k-1}W\left(\tfrac{3}{8}\pi(4k-3)\right).$ 9.9.10 $\displaystyle b_{k}$ $\displaystyle=-T\left(\tfrac{3}{8}\pi(4k-3)\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $b_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Bi}$, $k$: nonnegative integer and $T$: expansion Source: Olver (1954, (A 21)) Permalink: http://dlmf.nist.gov/9.9.E10 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9 9.9.11 $\displaystyle\mathrm{Bi}'\left(b_{k}\right)$ $\displaystyle=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-3)\right),$ 9.9.12 $\displaystyle b^{\prime}_{k}$ $\displaystyle=-U\left(\tfrac{3}{8}\pi(4k-1)\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $b^{\prime}_{\NVar{k}}$: $k$th zero of Airy $\mathrm{Bi}'$, $k$: nonnegative integer and $U$: expansion Source: Olver (1954, (A 21)) Permalink: http://dlmf.nist.gov/9.9.E12 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9 9.9.13 $\displaystyle\mathrm{Bi}\left(b^{\prime}_{k}\right)$ $\displaystyle=(-1)^{k}W\left(\tfrac{3}{8}\pi(4k-1)\right).$
 9.9.14 $\displaystyle\beta_{k}$ $\displaystyle=e^{\pi i/3}T\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{3}{4}i\ln 2\right),$ 9.9.15 $\displaystyle\mathrm{Bi}'\left(\beta_{k}\right)$ $\displaystyle=(-1)^{k}\sqrt{2}e^{-\pi i/6}V\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{% 3}{4}i\ln 2\right),$ 9.9.16 $\displaystyle\beta^{\prime}_{k}$ $\displaystyle=e^{\pi i/3}U\left(\tfrac{3}{8}\pi(4k-3)+\tfrac{3}{4}i\ln 2\right),$ 9.9.17 $\displaystyle\mathrm{Bi}\left(\beta^{\prime}_{k}\right)$ $\displaystyle=(-1)^{k-1}\sqrt{2}e^{\pi i/6}W\left(\tfrac{3}{8}\pi(4k-3)+\tfrac% {3}{4}i\ln 2\right).$

Here

 9.9.18 $T(t)\sim t^{2/3}\left(1+\frac{5}{48}t^{-2}-\frac{5}{36}t^{-4}+\frac{77125}{829% 44}t^{-6}-\frac{1080\;56875}{69\;67296}t^{-8}+\frac{16\;23755\;96875}{3344\;30% 208}t^{-10}-\cdots\right),$ ⓘ Defines: $T$: expansion (locally) Symbols: $\sim$: Poincaré asymptotic expansion Source: Olver (1954, (A 19)) Permalink: http://dlmf.nist.gov/9.9.E18 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9
 9.9.19 $U(t)\sim t^{2/3}\left(1-\frac{7}{48}t^{-2}+\frac{35}{288}t^{-4}-\frac{1\;81223% }{2\;07360}t^{-6}+\frac{186\;83371}{12\;44160}t^{-8}-\frac{9\;11458\;84361}{19% 11\;02976}t^{-10}+\cdots\right),$ ⓘ Defines: $U$: expansion (locally) Symbols: $\sim$: Poincaré asymptotic expansion Source: Olver (1954, (A 19)) Permalink: http://dlmf.nist.gov/9.9.E19 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9
 9.9.20 $V(t)\sim\pi^{-1/2}t^{1/6}\left(1+\frac{5}{48}t^{-2}-\frac{1525}{4608}t^{-4}+% \frac{23\;97875}{6\;63552}t^{-6}-\frac{7\;48989\;40625}{8918\;13888}t^{-8}+% \frac{14419\;83037\;34375}{4\;28070\;66624}t^{-10}-\cdots\right),$ ⓘ Defines: $V$: expansion (locally) Symbols: $\sim$: Poincaré asymptotic expansion and $\pi$: the ratio of the circumference of a circle to its diameter Source: Olver (1954, (A 19)) Permalink: http://dlmf.nist.gov/9.9.E20 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9
 9.9.21 $W(t)\sim\pi^{-1/2}t^{-1/6}\left(1-\frac{7}{96}t^{-2}+\frac{1673}{6144}t^{-4}-% \frac{843\;94709}{265\;42080}t^{-6}+\frac{78\;02771\;35421}{1\;01921\;58720}t^% {-8}-\frac{20444\;90510\;51945}{6\;52298\;15808}t^{-10}+\cdots\right).$ ⓘ Defines: $W$: expansion (locally) Symbols: $\sim$: Poincaré asymptotic expansion and $\pi$: the ratio of the circumference of a circle to its diameter Source: Olver (1954, (A 19)) Permalink: http://dlmf.nist.gov/9.9.E21 Encodings: TeX, pMML, png See also: Annotations for 9.9(iv), 9.9 and 9

For higher terms see Fabijonas and Olver (1999).

For error bounds for the asymptotic expansions of $a_{k}$, $b_{k}$, $a^{\prime}_{k}$, and $b^{\prime}_{k}$ 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 $\mathrm{Ai}$, $\mathrm{Ai}'$, $\mathrm{Bi}$, $\mathrm{Bi}'$, 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 $\mathrm{Bi}$ and $\mathrm{Bi}'$ in the upper half plane.