Airy functions

The Airy functions \(Ai(z)\), \(Bi(z)\) and their derivatives, for real or complex z, as scipy.special.airy() and airye() compute them. Each call returns all four, as a named list, as SciPy returns a tuple.

Usage

sc_airy(z)

sc_airye(z)

Arguments

z: Numeric or complex vector.

Value

A list with elements ai, aip, bi and bip, each a double vector, or a complex vector for complex z, with z's dimensions or names. NA in z gives NA in every element.

Details

sc_airy(): \(Ai\), \(Ai'\), \(Bi\), \(Bi'\).
sc_airye(): the exponentially scaled forms, \(Ai(z) e^{\zeta}\), \(Ai'(z) e^{\zeta}\), \(Bi(z) e^{-|\Re \zeta|}\) and \(Bi'(z) e^{-|\Re \zeta|}\), with \(\zeta = \frac23 z^{3/2}\).

Other packages

In gsl, for real arguments: airy_Ai(), airy_Bi(), airy_Ai_deriv(), airy_Bi_deriv() and their _scaled forms. In Bessel, for complex arguments: AiryA(z, deriv) and AiryB().

Examples

sc_airy(c(-1, 0, 1))
#> $ai
#> [1] 0.5355609 0.3550281 0.1352924
#> 
#> $aip
#> [1] -0.01016057 -0.25881940 -0.15914744
#> 
#> $bi
#> [1] 0.1039974 0.6149266 1.2074236
#> 
#> $bip
#> [1] 0.5923756 0.4482884 0.9324359
#> 
sc_airye(10)$ai
#> [1] 0.1581237
sc_airy(1 + 1i)
#> $ai
#> [1] 0.06045831-0.1518896i
#> 
#> $aip
#> [1] -0.130628+0.1630676i
#> 
#> $bi
#> [1] 0.7166581+0.6198893i
#> 
#> $bip
#> [1] 0.07566284+0.783701i
#>