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AiryAi

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

AiryAi[z]
    gives the Airy function Ai(z), the solution of y'' = z y that decays
    as z -> +Infinity.
AiryAi[0] = 1/(3^(2/3) Gamma[2/3]), AiryAi[+-Infinity] = 0. An entire
function of z. Real and complex inputs evaluate numerically at machine
or arbitrary (MPFR) precision; D[AiryAi[z], z] = AiryAiPrime[z]. Listable.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= AiryAi[0]
Out[1]= 1/(3^(2/3) Gamma[2/3])

In[2]:= AiryAi[1.8]
Out[2]= 0.0470362

Implementation notes

Attributes: Listable, NumericFunction, Protected, ReadProtected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= AiryAi[0]
Out[1]= 1/(3^(2/3) Gamma[2/3])

In[1]:= N[AiryAi[1], 40]
Out[1]= 0.13529241631288141552414742351546630617494

In[1]:= D[AiryAi[z], {z, 2}]
Out[1]= z AiryAi[z]

In[1]:= Series[AiryAi[z], {z, 0, 4}]
Out[1]= 1/(3^(2/3) Gamma[2/3]) + -1/(3^(1/3) Gamma[1/3]) z + 1/6/(3^(2/3) Gamma[2/3]) z^3 + -1/12/(3^(1/3) Gamma[1/3]) z^4 + O[z]^5

In[1]:= N[AiryAi[2] AiryBiPrime[2] - AiryAiPrime[2] AiryBi[2], 30]
Out[1]= 0.3183098861837906715377675267449

Notes

AiryAi[z] is the recessive solution of the Airy equation y'' == z y decaying as z -> +Infinity; the second-derivative identity D[AiryAi[z], {z, 2}] == z AiryAi[z] makes this explicit. The exact origin value is 1/(3^(2/3) Gamma[2/3]), and the Maclaurin series shows the characteristic missing z^2 term (every third coefficient vanishes). The last example is the Airy Wronskian Ai(z) Bi'(z) - Ai'(z) Bi(z) == 1/Pi, recovered numerically as 0.318309886.... Real and complex arguments evaluate at machine or MPFR precision; AiryAi is Listable.