§9.9 Zeros

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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
§9.9(ii) Relation to Modulus and Phase
§9.9(iii) Derivatives With Respect to
§9.9(iv) Asymptotic Expansions
§9.9(v) Tables

§9.9(i) Distribution and Notation

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Defines:: $\beta_{\NVar{k}}$ : th complex zero of Airy $\operatorname{Bi}$ , $\beta^{\prime}_{\NVar{k}}$ : th complex zero of Airy $\operatorname{Bi}'$ , $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $b_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}$ , $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ and $b^{\prime}_{\NVar{k}}$ : 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

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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 $\theta\left(a_{k}\right)=\phi\left(a^{\prime}_{k+1}\right)=k\pi,$

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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, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ and : 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 $\theta\left(b_{k}\right)=\phi\left(b^{\prime}_{k}\right)=(k-\tfrac{1}{2})\pi.$

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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, $b_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}$ , $b^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}'$ and : 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

$\operatorname{Ai}'\left(a_{k}\right)=\frac{(-1)^{k-1}}{\pi M\left(a_{k}\right)},$

$\operatorname{Bi}'\left(b_{k}\right)=\frac{(-1)^{k-1}}{\pi M\left(b_{k}\right)},$

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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, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $b_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}$ and : 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

$\operatorname{Ai}\left(a^{\prime}_{k}\right)=\frac{(-1)^{k-1}}{\pi N\left(a^{% \prime}_{k}\right)},$

$\operatorname{Bi}\left(b^{\prime}_{k}\right)=\frac{(-1)^{k}}{\pi N\left(b^{% \prime}_{k}\right)}.$

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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, $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ , $b^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}'$ and : 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

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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 is regarded as a continuous variable, then

9.9.5

$\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}\left(-\frac{\mathrm{d}a_{k}}{% \mathrm{d}k}\right)^{-1/2},$

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ and : 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

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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

9.9.6 $a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , : nonnegative integer and : 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 $\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)% \right),$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , : nonnegative integer and : 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 $a^{\prime}_{k}=-U\left(\tfrac{3}{8}\pi(4k-3)\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ , : nonnegative integer and : 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 $\operatorname{Ai}\left(a^{\prime}_{k}\right)=(-1)^{k-1}W\left(\tfrac{3}{8}\pi(% 4k-3)\right).$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ , : nonnegative integer and : 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 $b_{k}=-T\left(\tfrac{3}{8}\pi(4k-3)\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $b_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}$ , : nonnegative integer and : 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 $\operatorname{Bi}'\left(b_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-3)% \right),$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $b_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}$ , : nonnegative integer and : 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 $b^{\prime}_{k}=-U\left(\tfrac{3}{8}\pi(4k-1)\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $b^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}'$ , : nonnegative integer and : 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 $\operatorname{Bi}\left(b^{\prime}_{k}\right)=(-1)^{k}W\left(\tfrac{3}{8}\pi(4k% -1)\right).$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $b^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}'$ , : nonnegative integer and : 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 $\beta_{k}=e^{\pi i/3}T\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{3}{4}i\ln 2\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta_{\NVar{k}}$ : 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, : nonnegative integer and : 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 $\operatorname{Bi}'\left(\beta_{k}\right)=(-1)^{k}\sqrt{2}e^{-\pi i/6}V\left(% \tfrac{3}{8}\pi(4k-1)+\tfrac{3}{4}i\ln 2\right),$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta_{\NVar{k}}$ : 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, : nonnegative integer and : 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 $\beta^{\prime}_{k}=e^{\pi i/3}U\left(\tfrac{3}{8}\pi(4k-3)+\tfrac{3}{4}i\ln 2% \right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta^{\prime}_{\NVar{k}}$ : 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, : nonnegative integer and : 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 $\operatorname{Bi}\left(\beta^{\prime}_{k}\right)=(-1)^{k-1}\sqrt{2}e^{\pi i/6}% W\left(\tfrac{3}{8}\pi(4k-3)+\tfrac{3}{4}i\ln 2\right).$

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta^{\prime}_{\NVar{k}}$ : 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, : nonnegative integer and : 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 $T(t)\sim t^{2/3}\left(1+\frac{5}{48}t^{-2}-\frac{5}{36}t^{-4}+\frac{77125}{829% 44}t^{-6}-\frac{1080\;56875}{69\;67296}t^{-8}+\frac{16\;23755\;96875}{3344\;30% 208}t^{-10}-\cdots\right),$

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Defines:: : 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 $U(t)\sim t^{2/3}\left(1-\frac{7}{48}t^{-2}+\frac{35}{288}t^{-4}-\frac{1\;81223% }{2\;07360}t^{-6}+\frac{186\;83371}{12\;44160}t^{-8}-\frac{9\;11458\;84361}{19% 11\;02976}t^{-10}+\cdots\right),$

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Defines:: : 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:: : 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:: : 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

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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 $\operatorname{Ai}'$ .

	$a_{k}$	$\operatorname{Ai}'\left(a_{k}\right)$	$a^{\prime}_{k}$	$\operatorname{Ai}\left(a^{\prime}_{k}\right)$
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, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ and : 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 $\operatorname{Bi}'$ .

	$b_{k}$	$\operatorname{Bi}'\left(b_{k}\right)$	$b^{\prime}_{k}$	$\operatorname{Bi}\left(b^{\prime}_{k}\right)$
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, $b_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}$ , $b^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}'$ and : 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 $\operatorname{Bi}$ .

	$e^{-\pi i/3}\beta_{k}$		$\operatorname{Bi}'\left(\beta_{k}\right)$
	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, $\beta_{\NVar{k}}$ : th complex zero of Airy $\operatorname{Bi}$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : 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 $\operatorname{Bi}'$ .

	$e^{-\pi i/3}\beta^{\prime}_{k}$		$\operatorname{Bi}\left(\beta^{\prime}_{k}\right)$
	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

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\beta^{\prime}_{\NVar{k}}$ : th complex zero of Airy $\operatorname{Bi}'$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : nonnegative integer
Proof sketch:: For computations see §9.17(v).
Referenced by:: §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T4
See also:: Annotations for §9.9(v), §9.9 and Ch.9

9.8 Modulus and Phase 9.10 Integrals

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