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Airy functions

Table of contents: Differential equation - Special values - Higher derivatives - Hypergeometric representations - Analytic properties

Differential equation

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y(z)zy ⁣(z)=0   where y ⁣(z)=CAi ⁣(z)+DBi ⁣(z)y''(z) - z y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)
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Ai ⁣(z)Bi ⁣(z)Ai ⁣(z)Bi ⁣(z)=1π\operatorname{Ai}\!\left(z\right) \operatorname{Bi}'\!\left(z\right) - \operatorname{Ai}'\!\left(z\right) \operatorname{Bi}\!\left(z\right) = \frac{1}{\pi}

Special values

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Ai ⁣(0)=132/3Γ ⁣(23)[0.355028053887817239260063186004±1.841031]\operatorname{Ai}\!\left(0\right) = \frac{1}{{3}^{2 / 3} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.355028053887817239260063186004 \pm 1.84 \cdot 10^{-31}\right]
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Ai ⁣(0)=131/3Γ ⁣(13)[0.258819403792806798405183560189±2.041031]\operatorname{Ai}'\!\left(0\right) = -\frac{1}{{3}^{1 / 3} \Gamma\!\left(\frac{1}{3}\right)} \in \left[-0.258819403792806798405183560189 \pm 2.04 \cdot 10^{-31}\right]
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Bi ⁣(0)=131/6Γ ⁣(23)[0.614926627446000735150922369094±3.871031]\operatorname{Bi}\!\left(0\right) = \frac{1}{{3}^{1 / 6} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.614926627446000735150922369094 \pm 3.87 \cdot 10^{-31}\right]
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Bi ⁣(0)=31/6Γ ⁣(13)[0.448288357353826357914823710399±1.721031]\operatorname{Bi}'\!\left(0\right) = \frac{{3}^{1 / 6}}{\Gamma\!\left(\frac{1}{3}\right)} \in \left[0.448288357353826357914823710399 \pm 1.72 \cdot 10^{-31}\right]

Higher derivatives

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Ai(z)=zAi ⁣(z)\operatorname{Ai}''(z) = z \operatorname{Ai}\!\left(z\right)
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Bi(z)=zBi ⁣(z)\operatorname{Bi}''(z) = z \operatorname{Bi}\!\left(z\right)
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y(n)(z)=zy(n2)(z)+(n2)y(n3)(z)   where y ⁣(z)=CAi ⁣(z)+DBi ⁣(z){y}^{(n)}(z) = z {y}^{(n - 2)}(z) + \left(n - 2\right) {y}^{(n - 3)}(z)\; \text{ where } y\!\left(z\right) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)

Hypergeometric representations

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Ai ⁣(z)=Ai ⁣(0)0F1 ⁣(23,z39)+zAi ⁣(0)0F1 ⁣(43,z39)\operatorname{Ai}\!\left(z\right) = \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)
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Bi ⁣(z)=Bi ⁣(0)0F1 ⁣(23,z39)+zBi ⁣(0)0F1 ⁣(43,z39)\operatorname{Bi}\!\left(z\right) = \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)
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Ai ⁣(z)=Ai ⁣(0)0F1 ⁣(13,z39)+z22Ai ⁣(0)0F1 ⁣(53,z39)\operatorname{Ai}'\!\left(z\right) = \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)
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Bi ⁣(z)=Bi ⁣(0)0F1 ⁣(13,z39)+z22Bi ⁣(0)0F1 ⁣(53,z39)\operatorname{Bi}'\!\left(z\right) = \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)

Analytic properties

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HolomorphicDomain ⁣(CAi ⁣(z)+DBi ⁣(z),z,C{~})=C\operatorname{HolomorphicDomain}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C}
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Poles ⁣(CAi ⁣(z)+DBi ⁣(z),z,C{~})={}\operatorname{Poles}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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EssentialSingularities ⁣(CAi ⁣(z)+DBi ⁣(z),z,C{~})={~}\operatorname{EssentialSingularities}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
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BranchPoints ⁣(CAi ⁣(z)+DBi ⁣(z),z,C{~})={}\operatorname{BranchPoints}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchCuts ⁣(CAi ⁣(z)+DBi ⁣(z),z,C)={}\operatorname{BranchCuts}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}
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zeroszCAi ⁣(z)R\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}\!\left(z\right) \subset \mathbb{R}
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zeroszCAi ⁣(z)R\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}'\!\left(z\right) \subset \mathbb{R}

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-05-23 08:00:13.607731 UTC