§9.9 Zeros

ⓘ

Referenced by:: §10.21(vii), §10.21(viii), Ch.9
Permalink:: http://dlmf.nist.gov/9.9
See also:: Annotations for Ch.9

§9.9(i) Distribution and Notation

ⓘ

Defines:: $\beta_{\NVar{k}}$ : $k$ th complex zero of Airy $\operatorname{Bi}$ , $\beta^{\prime}_{\NVar{k}}$ : $k$ th complex zero of Airy $\operatorname{Bi}'$ , $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $b_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}$ , $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ and $b^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}'$
Notes:: See Olver (1997b, pp. 414–415).
Referenced by:: §12.11(iii), §18.16(iv), §18.16(v), §18.24, Erratum (V1.0.11) for References, Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/9.9.i
Addition (effective with 1.0.11):: A reference to Gil and Segura (2014) was added.
See also:: Annotations for §9.9 and Ch.9

On the real line, $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ 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 $k=1,2,\ldots.$

$\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ have no other zeros. However, $\operatorname{Bi}\left(z\right)$ and $\operatorname{Bi}'\left(z\right)$ 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 $\overline{\beta_{k}}$ , $\overline{\beta^{\prime}_{k}}$ , 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 $\operatorname{Ai}'\left(z\right)-\sigma\operatorname{Ai}\left(z\right)$ , where $\sigma$ is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014).

§9.9(ii) Relation to Modulus and Phase

ⓘ

Keywords:: Airy functions, modulus and phase, relation to modulus and phase, relation to zeros, zeros
Notes:: See Miller (1946, p. B48) and Olver (1997b, p. 404).
Permalink:: http://dlmf.nist.gov/9.9.ii
See also:: Annotations for §9.9 and Ch.9

9.9.1	$\displaystyle\theta\left(a_{k}\right)$	$\displaystyle=\phi\left(a^{\prime}_{k+1}\right)=k\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\phi\left(\NVar{z}\right)$ : Airy phase function, $\theta\left(\NVar{z}\right)$ : Airy phase function, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ and $k$ : nonnegative integer Source: Olver (1997b, (5.04), p. 404) Referenced by: (9.9.3), (9.9.4) Permalink: http://dlmf.nist.gov/9.9.E1 Encodings: TeX, pMML, png See also: Annotations for §9.9(ii), §9.9 and Ch.9
9.9.2	$\displaystyle\theta\left(b_{k}\right)$	$\displaystyle=\phi\left(b^{\prime}_{k}\right)=(k-\tfrac{1}{2})\pi.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\phi\left(\NVar{z}\right)$ : Airy phase function, $\theta\left(\NVar{z}\right)$ : Airy phase function, $b_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}$ , $b^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}'$ and $k$ : nonnegative integer Source: Olver (1997b, (5.04), p. 404) Referenced by: (9.9.3), (9.9.4) Permalink: http://dlmf.nist.gov/9.9.E2 Encodings: TeX, pMML, png See also: Annotations for §9.9(ii), §9.9 and Ch.9

9.9.3	$\displaystyle\operatorname{Ai}'\left(a_{k}\right)$	$\displaystyle=\frac{(-1)^{k-1}}{\pi M\left(a_{k}\right)},$
9.9.3	$\displaystyle\operatorname{Bi}'\left(b_{k}\right)$	$\displaystyle=\frac{(-1)^{k-1}}{\pi M\left(b_{k}\right)},$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : 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 $\operatorname{Ai}$ , $b_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}$ and $k$ : nonnegative integer Proof sketch: 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 Ch.9

9.9.4	$\displaystyle\operatorname{Ai}\left(a^{\prime}_{k}\right)$	$\displaystyle=\frac{(-1)^{k-1}}{\pi N\left(a^{\prime}_{k}\right)},$
9.9.4	$\displaystyle\operatorname{Bi}\left(b^{\prime}_{k}\right)$	$\displaystyle=\frac{(-1)^{k}}{\pi N\left(b^{\prime}_{k}\right)}.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : 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 $\operatorname{Ai}'$ , $b^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}'$ and $k$ : nonnegative integer Proof sketch: 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 Ch.9

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

ⓘ

Keywords:: Airy functions, differentiation, zeros
Referenced by:: §9.1
Permalink:: http://dlmf.nist.gov/9.9.iii
See also:: Annotations for §9.9 and Ch.9

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

9.9.5	$\displaystyle\operatorname{Ai}'\left(a_{k}\right)$	$\displaystyle=(-1)^{k-1}\left(-\frac{\mathrm{d}a_{k}}{\mathrm{d}k}\right)^{-1/% 2},$
9.9.5	$\displaystyle\operatorname{Ai}\left(a^{\prime}_{k}\right)$	$\displaystyle=(-1)^{k-1}\left(a^{\prime}_{k}\frac{\mathrm{d}a^{\prime}_{k}}{% \mathrm{d}k}\right)^{-1/2}.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ and $k$ : nonnegative integer Source: Olver (1954, (A 23)) Permalink: http://dlmf.nist.gov/9.9.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §9.9(iii), §9.9 and Ch.9

See Olver (1954, Appendix).

§9.9(iv) Asymptotic Expansions

ⓘ

Keywords:: Airy functions, asymptotic expansions, zeros
Notes:: See Olver (1954, Appendix).
Referenced by:: §9.17(v)
Permalink:: http://dlmf.nist.gov/9.9.iv
See also:: Annotations for §9.9 and Ch.9

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 $\operatorname{Ai}$ , $k$ : 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 Ch.9
9.9.7	$\displaystyle\operatorname{Ai}'\left(a_{k}\right)$	$\displaystyle=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $V$ : expansion Source: Olver (1954, (A 20)) Permalink: http://dlmf.nist.gov/9.9.E7 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9
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 $\operatorname{Ai}'$ , $k$ : 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 Ch.9
9.9.9	$\displaystyle\operatorname{Ai}\left(a^{\prime}_{k}\right)$	$\displaystyle=(-1)^{k-1}W\left(\tfrac{3}{8}\pi(4k-3)\right).$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ , $k$ : nonnegative integer and $W$ : expansion Source: Olver (1954, (A 20)) Permalink: http://dlmf.nist.gov/9.9.E9 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9
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 $\operatorname{Bi}$ , $k$ : 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 Ch.9
9.9.11	$\displaystyle\operatorname{Bi}'\left(b_{k}\right)$	$\displaystyle=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-3)\right),$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $b_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}$ , $k$ : nonnegative integer and $V$ : expansion Source: Olver (1954, (A 21)) Permalink: http://dlmf.nist.gov/9.9.E11 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9
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 $\operatorname{Bi}'$ , $k$ : 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 Ch.9
9.9.13	$\displaystyle\operatorname{Bi}\left(b^{\prime}_{k}\right)$	$\displaystyle=(-1)^{k}W\left(\tfrac{3}{8}\pi(4k-1)\right).$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $b^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}'$ , $k$ : nonnegative integer and $W$ : expansion Source: Olver (1954, (A 21)) Permalink: http://dlmf.nist.gov/9.9.E13 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9

9.9.14	$\displaystyle\beta_{k}$	$\displaystyle=e^{\pi i/3}T\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{3}{4}i\ln 2\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta_{\NVar{k}}$ : $k$ th complex zero of Airy $\operatorname{Bi}$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $T$ : expansion Source: Olver (1954, (A 22)) Permalink: http://dlmf.nist.gov/9.9.E14 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9
9.9.15	$\displaystyle\operatorname{Bi}'\left(\beta_{k}\right)$	$\displaystyle=(-1)^{k}\sqrt{2}e^{-\pi i/6}V\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{% 3}{4}i\ln 2\right),$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta_{\NVar{k}}$ : $k$ th complex zero of Airy $\operatorname{Bi}$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $V$ : expansion Source: Olver (1954, (A 22)) Permalink: http://dlmf.nist.gov/9.9.E15 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9
9.9.16	$\displaystyle\beta^{\prime}_{k}$	$\displaystyle=e^{\pi i/3}U\left(\tfrac{3}{8}\pi(4k-3)+\tfrac{3}{4}i\ln 2\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta^{\prime}_{\NVar{k}}$ : $k$ th complex zero of Airy $\operatorname{Bi}'$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $U$ : expansion Source: Olver (1954, (A 22)) Permalink: http://dlmf.nist.gov/9.9.E16 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9
9.9.17	$\displaystyle\operatorname{Bi}\left(\beta^{\prime}_{k}\right)$	$\displaystyle=(-1)^{k-1}\sqrt{2}e^{\pi i/6}W\left(\tfrac{3}{8}\pi(4k-3)+\tfrac% {3}{4}i\ln 2\right).$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta^{\prime}_{\NVar{k}}$ : $k$ th complex zero of Airy $\operatorname{Bi}'$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $W$ : expansion Source: Olver (1954, (A 22)) Permalink: http://dlmf.nist.gov/9.9.E17 Encodings: TeX, pMML, png See also: Annotations for §9.9(iv), §9.9 and Ch.9

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 Ch.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 Ch.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 Ch.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 Ch.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

ⓘ

Keywords:: Airy functions, tables, zeros
Referenced by:: §9.17(v), 5th item
Permalink:: http://dlmf.nist.gov/9.9.v
See also:: Annotations for §9.9 and Ch.9

Tables 9.9.1 and 9.9.2 give 10D values of the first ten real zeros of $\operatorname{Ai}$ , $\operatorname{Ai}'$ , $\operatorname{Bi}$ , $\operatorname{Bi}'$ , 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 $\operatorname{Bi}$ and $\operatorname{Bi}'$ in the upper half plane.

Table 9.9.1: Zeros of

\operatorname{Ai}

and

\operatorname{Ai}'

$k$	$a_{k}$	$\operatorname{Ai}'\left(a_{k}\right)$	$a^{\prime}_{k}$	$\operatorname{Ai}\left(a^{\prime}_{k}\right)$
$1$	$-2.33810\;74105$	$0.70121\;08227$	$-1.01879\;29716$	$0.53565\;66560$
$2$	$-4.08794\;94441$	$-0.80311\;13697$	$-3.24819\;75822$	$-0.41901\;54780$
$3$	$-5.52055\;98281$	$0.86520\;40259$	$-4.82009\;92112$	$0.38040\;64686$
$4$	$-6.78670\;80901$	$-0.91085\;07370$	$-6.16330\;73556$	$-0.35790\;79437$
$5$	$-7.94413\;35871$	$0.94733\;57094$	$-7.37217\;72550$	$0.34230\;12444$
$6$	$-9.02265\;08533$	$-0.97792\;28086$	$-8.48848\;67340$	$-0.33047\;62291$
$7$	$-10.04017\;43416$	$1.00437\;01227$	$-9.53544\;90524$	$0.32102\;22882$
$8$	$-11.00852\;43037$	$-1.02773\;86888$	$-10.52766\;03970$	$-0.31318\;53910$
$9$	$-11.93601\;55632$	$1.04872\;06486$	$-11.47505\;66335$	$0.30651\;72939$
$10$	$-12.82877\;67529$	$-1.06779\;38592$	$-12.38478\;83718$	$-0.30073\;08293$

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ and $k$ : nonnegative integer
Proof sketch:: For computations see §9.17(v).
Referenced by:: §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T1
See also:: Annotations for §9.9(v), §9.9 and Ch.9

Table 9.9.2: Real zeros of

\operatorname{Bi}

and

\operatorname{Bi}'

$k$	$b_{k}$	$\operatorname{Bi}'\left(b_{k}\right)$	$b^{\prime}_{k}$	$\operatorname{Bi}\left(b^{\prime}_{k}\right)$
$1$	$-1.17371\;32227$	$0.60195\;78880$	$-2.29443\;96826$	$-0.45494\;43836$
$2$	$-3.27109\;33028$	$-0.76031\;01415$	$-4.07315\;50891$	$0.39652\;28361$
$3$	$-4.83073\;78417$	$0.83699\;10126$	$-5.51239\;57297$	$-0.36796\;91615$
$4$	$-6.16985\;21283$	$-0.88947\;99014$	$-6.78129\;44460$	$0.34949\;91168$
$5$	$-7.37676\;20794$	$0.92998\;36386$	$-7.94017\;86892$	$-0.33602\;62401$
$6$	$-8.49194\;88465$	$-0.96323\;44302$	$-9.01958\;33588$	$0.32550\;97364$
$7$	$-9.53819\;43793$	$0.99158\;63705$	$-10.03769\;63349$	$-0.31693\;46537$
$8$	$-10.52991\;35067$	$-1.01638\;96592$	$-11.00646\;26677$	$0.30972\;59408$
$9$	$-11.47695\;35513$	$1.03849\;42860$	$-11.93426\;16450$	$-0.30352\;76648$
$10$	$-12.38641\;71386$	$-1.05847\;18444$	$-12.82725\;83092$	$0.29810\;49111$

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $b_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}$ , $b^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Bi}'$ and $k$ : nonnegative integer
Proof sketch:: For computations see §9.17(v).
Referenced by:: §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T2
See also:: Annotations for §9.9(v), §9.9 and Ch.9

Table 9.9.3: Complex zeros of

\operatorname{Bi}

	$e^{-\pi i/3}\beta_{k}$		$\operatorname{Bi}'\left(\beta_{k}\right)$
$k$	modulus	phase	modulus	phase
$1$	$2.35387\;33809$	$0.09533\;49591$	$0.99310\;68457$	$2.64060\;02521$
$2$	$4.09328\;73094$	$0.04178\;55604$	$1.13612\;83345$	$-0.51328\;28720$
$3$	$5.52350\;35011$	$0.02668\;05442$	$1.22374\;37881$	$2.62462\;83591$
$4$	$6.78865\;95301$	$0.01958\;69751$	$1.28822\;92493$	$-0.51871\;63829$
$5$	$7.94555\;90160$	$0.01547\;08228$	$1.33979\;47726$	$2.62185\;44560$
$6$	$9.02375\;63663$	$0.01278\;34808$	$1.38303\;39005$	$-0.52040\;69437$
$7$	$10.04106\;73680$	$0.01089\;12610$	$1.42042\;53456$	$2.62071\;41895$
$8$	$11.00926\;72579$	$0.00948\;68445$	$1.45346\;64633$	$-0.52122\;87219$
$9$	$11.93664\;76131$	$0.00840\;31785$	$1.48313\;45656$	$2.62009\;35195$
$10$	$12.82932\;39388$	$0.00754\;16607$	$1.51010\;46383$	$-0.52171\;41947$

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta_{\NVar{k}}$ : $k$ th complex zero of Airy $\operatorname{Bi}$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $k$ : nonnegative integer
Proof sketch:: For computations see §9.17(v).
Referenced by:: §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T3
See also:: Annotations for §9.9(v), §9.9 and Ch.9

Table 9.9.4: Complex zeros of

\operatorname{Bi}'

	$e^{-\pi i/3}\beta^{\prime}_{k}$		$\operatorname{Bi}\left(\beta^{\prime}_{k}\right)$
$k$	modulus	phase	modulus	phase
$1$	$1.12139\;32942$	$0.33072\;66208$	$0.75004\;14897$	$0.46597\;78930$
$2$	$3.25690\;82266$	$0.05938\;99367$	$0.59221\;66315$	$-2.63235\;40329$
$3$	$4.82400\;26102$	$0.03278\;56423$	$0.53787\;06321$	$0.51549\;32992$
$4$	$6.16568\;66408$	$0.02266\;24588$	$0.50611\;02160$	$-2.62362\;85920$
$5$	$7.37383\;79870$	$0.01731\;96481$	$0.48406\;00643$	$0.51928\;28169$
$6$	$8.48973\;85596$	$0.01401\;65283$	$0.46734\;68449$	$-2.62149\;05716$
$7$	$9.53644\;07072$	$0.01177\;19311$	$0.45398\;23240$	$0.52066\;02139$
$8$	$10.52847\;37502$	$0.01014\;71783$	$0.44290\;25018$	$-2.62052\;78353$
$9$	$11.47574\;11237$	$0.00891\;66153$	$0.43347\;44668$	$0.52137\;15495$
$10$	$12.38537\;59341$	$0.00795\;22843$	$0.42529\;25837$	$-2.61998\;05803$