# §9.9 Zeros

## §9.9(i) Distribution and Notation

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

$\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)$ have no other zeros. However, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)$ each have an infinite number of complex zeros. They lie in the sectors $\tfrac{1}{3}\pi<\mathop{\mathrm{ph}\/}\nolimits z<\tfrac{1}{2}\pi$ and $-\tfrac{1}{2}\pi<\mathop{\mathrm{ph}\/}\nolimits z<-\tfrac{1}{3}\pi$, and are denoted by $\mathop{\beta_{k}\/}\nolimits$, $\mathop{\beta^{\prime}_{k}\/}\nolimits$, respectively, in the former sector, and by $\bar{\mathop{\beta_{k}\/}\nolimits}$, $\bar{\mathop{\beta^{\prime}_{k}\/}\nolimits}$, 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 $\Complex$ of the zeros of $\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)-\sigma\mathop{\mathrm{Ai}\/}% \nolimits\!\left(z\right)$, where $\sigma$ is an arbitrary complex constant, see Muraveĭ (1976).

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

 9.9.1 $\displaystyle\mathop{\theta\/}\nolimits\!\left(\mathop{a_{k}\/}\nolimits\right)$ $\displaystyle=\mathop{\phi\/}\nolimits\!\left(\mathop{a^{\prime}_{k+1}\/}% \nolimits\right)=k\pi,$ 9.9.2 $\displaystyle\mathop{\theta\/}\nolimits\!\left(\mathop{b_{k}\/}\nolimits\right)$ $\displaystyle=\mathop{\phi\/}\nolimits\!\left(\mathop{b^{\prime}_{k}\/}% \nolimits\right)=(k-\tfrac{1}{2})\pi.$
 9.9.3 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(\mathop{a_{k}\/}\nolimits\right)$ $\displaystyle=\frac{(-1)^{k-1}}{\pi\mathop{M\/}\nolimits\!\left(\mathop{a_{k}% \/}\nolimits\right)},$ $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(\mathop{b_{k}\/}\nolimits\right)$ $\displaystyle=\frac{(-1)^{k-1}}{\pi\mathop{M\/}\nolimits\!\left(\mathop{b_{k}% \/}\nolimits\right)},$
 9.9.4 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(\mathop{a^{\prime}_{k}\/}% \nolimits\right)$ $\displaystyle=\frac{(-1)^{k-1}}{\pi\mathop{N\/}\nolimits\!\left(\mathop{a^{% \prime}_{k}\/}\nolimits\right)},$ $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(\mathop{b^{\prime}_{k}\/}% \nolimits\right)$ $\displaystyle=\frac{(-1)^{k}}{\pi\mathop{N\/}\nolimits\!\left(\mathop{b^{% \prime}_{k}\/}\nolimits\right)}.$

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

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

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

See Olver (1954, Appendix).

## §9.9(iv) Asymptotic Expansions

For large $k$

 9.9.6 $\displaystyle\mathop{a_{k}\/}\nolimits$ $\displaystyle=-T\left(\tfrac{3}{8}\pi(4k-1)\right),$ 9.9.7 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(\mathop{a_{k}\/}\nolimits\right)$ $\displaystyle=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)\right),$ 9.9.8 $\displaystyle\mathop{a^{\prime}_{k}\/}\nolimits$ $\displaystyle=-U\left(\tfrac{3}{8}\pi(4k-3)\right),$ 9.9.9 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(\mathop{a^{\prime}_{k}\/}% \nolimits\right)$ $\displaystyle=(-1)^{k-1}W\left(\tfrac{3}{8}\pi(4k-3)\right).$ 9.9.10 $\displaystyle\mathop{b_{k}\/}\nolimits$ $\displaystyle=-T\left(\tfrac{3}{8}\pi(4k-3)\right),$ 9.9.11 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(\mathop{b_{k}\/}\nolimits\right)$ $\displaystyle=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-3)\right),$ 9.9.12 $\displaystyle\mathop{b^{\prime}_{k}\/}\nolimits$ $\displaystyle=-U\left(\tfrac{3}{8}\pi(4k-1)\right),$ 9.9.13 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(\mathop{b^{\prime}_{k}\/}% \nolimits\right)$ $\displaystyle=(-1)^{k}W\left(\tfrac{3}{8}\pi(4k-1)\right).$
 9.9.14 $\displaystyle\mathop{\beta_{k}\/}\nolimits$ $\displaystyle=e^{\pi i/3}T\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{3}{4}i\mathop{\ln% \/}\nolimits 2\right),$ 9.9.15 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(\mathop{\beta_{k}\/}% \nolimits\right)$ $\displaystyle=(-1)^{k}\sqrt{2}e^{-\pi i/6}V\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{% 3}{4}i\mathop{\ln\/}\nolimits 2\right),$ 9.9.16 $\displaystyle\mathop{\beta^{\prime}_{k}\/}\nolimits$ $\displaystyle=e^{\pi i/3}U\left(\tfrac{3}{8}\pi(4k-3)+\tfrac{3}{4}i\mathop{\ln% \/}\nolimits 2\right),$ 9.9.17 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(\mathop{\beta^{\prime}_{k}% \/}\nolimits\right)$ $\displaystyle=(-1)^{k-1}\sqrt{2}e^{\pi i/6}W\left(\tfrac{3}{8}\pi(4k-3)+\tfrac% {3}{4}i\mathop{\ln\/}\nolimits 2\right).$

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: $\sim$: asymptotic equality Permalink: http://dlmf.nist.gov/9.9.E18 Encodings: TeX, pMML, png
 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: $\sim$: asymptotic equality Permalink: http://dlmf.nist.gov/9.9.E19 Encodings: TeX, pMML, png
 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: $\sim$: asymptotic equality Permalink: http://dlmf.nist.gov/9.9.E20 Encodings: TeX, pMML, png
 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: $\sim$: asymptotic equality Permalink: http://dlmf.nist.gov/9.9.E21 Encodings: TeX, pMML, png

For higher terms see Fabijonas and Olver (1999).

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