AiryBi

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Airy function of the second kind

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaAiry function↗Airy function of the second kind↗MathWorldAiryFunctions↗AiryBi↗WikidataQ109729257↗DLMF9.2.i↗FungrimAiryBi↗Fungrim entryWolfram LanguageAiryBi✓↗

AiryBi(complex | infinity) -> numberAs native, and keeps its digits past a double's for real argument (N(AiryBi(2), 50)).

SageMathairy_bi(complex | infinity)mpmathairybi(complex | infinity)Juliaenumeratio_ball(airy_bi, complex | infinity)

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity) -> number	compute-engine	always

Domain: Compute engine

Plot

the complex plot cannot lower AiryBi

Examples

Basic #

$\mathrm{Bi}(0) = \sqrt{3}\,\mathrm{Ai}(0)$ numerically

live ↗

In$N(\operatorname{Bi}(0))$Out$0.614\,926\,627\,446\,000\,7$

Known value · mpmath 1.3.0 airybi

live ↗

In$\operatorname{Series}(\operatorname{Bi}(x), x, 0, 4)$Out$\frac{1}{Γ(\frac{2}{3})\sqrt[6]{3}}+\frac{x\sqrt[6]{3}}{Γ(\frac{1}{3})}+\frac{x^3}{6Γ(\frac{2}{3})\sqrt[6]{3}}+\frac{x^4\sqrt[6]{3}}{12Γ(\frac{1}{3})}+O\left(x^6\right)$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\operatorname{Bi}(x), x, ∞, 2))$Out$(\frac{5}{48\sqrt{π}x^{\frac{7}{4}}}+\frac{1}{\sqrt{π}\sqrt[4]{x}})e^{\frac{1}{3}×2x^{3/2}}$

Scope #

live ↗

In$N(\operatorname{Bi}(2), 50)$Out$3.298\,094\,999\,978\,214\,710\,280\,604\,425\,223\,452\,422\,003\,975\,963\,403\,6$

live ↗

In$N(\mathrm{FunctionRange}(\operatorname{Bi}(x), x, y))$Out$-0.454\,944\,383\,639\,657\,33≤y$

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In$\frac{\mathrm{d}\operatorname{Bi}(x)}{\mathrm{d}x}$Out$\operatorname{Bi}'(x)$

live ↗

In$\operatorname{Series}(\operatorname{Bi}(x), x, 0, 7)$Out$\frac{1}{Γ(\frac{2}{3})\sqrt[6]{3}}+\frac{x\sqrt[6]{3}}{Γ(\frac{1}{3})}+\frac{x^3}{6Γ(\frac{2}{3})\sqrt[6]{3}}+\frac{x^4\sqrt[6]{3}}{12Γ(\frac{1}{3})}+\frac{x^6}{180Γ(\frac{2}{3})\sqrt[6]{3}}+\frac{x^7\sqrt[6]{3}}{504Γ(\frac{1}{3})}+O\left(x^9\right)$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\operatorname{Bi}(x), x, \mathrm{DirectedInfinity}(z), 1))$Out$\operatorname{Series}(\operatorname{Bi}(x), x, \mathrm{DirectedInfinity}(z), 1)$

live ↗

In$\mathrm{Simplify}(\mathrm{FunctionExpand}(\operatorname{Bi}(ze^{(2πi)/3})))$Out$\frac{1}{4}(\sqrt{3}-i)(3\operatorname{Ai}(z)+i\operatorname{Bi}(z))$

About

Examples	9
Checked against	Wolfram Language 8 agree1 unevaluatedSageMath 5 agree1 unevaluatedmpmath 2 agreeJulia 2 agree
Crosswalk	Wikipedia ×2MathWorld ×2WikidataDLMFFungrimFungrim entry ×9Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts