Airy {Bessel}

Airy Functions (and Their First Derivative)

Description

Compute the Airy functions Ai or Bi or their first derivatives, \frac{d}{dz} Ai(z) and \frac{d}{dz} Bi(z).

The Airy functions are solutions of the differential equation

w'' = z w

for w(z), and are related to each other and to the (modified) Bessel functions via (many identities, see https://dlmf.nist.gov/9.6), e.g., if \zeta := \frac{2}{3} z \sqrt{z} = \frac{2}{3} z^{\frac{3}{2}},

Ai(z) = \pi^{-1}\sqrt{z/3}K_{1/3}(\zeta) = \frac{1}{3}\sqrt{z}\left(I_{-1/3}(\zeta) - I_{1/3}(\zeta)\right),

and

Bi(z) = \sqrt{z/3} \left(I_{-1/3}(\zeta) + I_{1/3}(\zeta)\right).

Usage

AiryA(z, deriv = 0, expon.scaled = FALSE, verbose = 0)
AiryB(z, deriv = 0, expon.scaled = FALSE, verbose = 0)

Arguments

z

complex or numeric vector.

deriv

order of derivative; must be 0 or 1.

expon.scaled

logical indicating if the result should be scaled by an exponential factor (typically to avoid under- or over-flow).

verbose

integer defaulting to 0, indicating the level of verbosity notably from C code.

Details

By default, when expon.scaled is false, AiryA() computes the complex Airy function Ai(z) or its derivative \frac{d}{dz} Ai(z) on deriv=0 or deriv=1 respectively.
When expon.scaled is true, it returns \exp(\zeta) Ai(z) or \exp(\zeta) \frac{d}{dz} Ai(z), effectively removing the exponential decay in -\pi/3 < \arg(z) < \pi/3 and the exponential growth in \pi/3 < \left|\arg(z)\right| < \pi, where \zeta= \frac{2}{3} z \sqrt{z}, and \arg(z) = Arg(z).

While the Airy functions Ai(z) and d/dz Ai(z) are analytic in the whole z plane, the corresponding scaled functions (for expon.scaled=TRUE) have a cut along the negative real axis.

By default, when expon.scaled is false, AiryB() computes the complex Airy function Bi(z) or its derivative \frac{d}{dz} Bi(z) on deriv=0 or deriv=1 respectively.
When expon.scaled is true, it returns exp(-\left|\Re(\zeta)\right|) Bi(z) or exp(-\left|\Re(\zeta)\right|)\frac{d}{dz}Bi(z), to remove the exponential behavior in both the left and right half planes where, as above, \zeta= \frac{2}{3}\cdot z \sqrt{z}.

Value

a complex or numeric vector of the same length (and class) as z.

Author(s)

Donald E. Amos, Sandia National Laboratories, wrote the original fortran code. Martin Maechler did the R interface.

References

see BesselJ; notably for many results the

Digital Library of Mathematical Functions (DLMF), Chapter 9 Airy and Related Functions at https://dlmf.nist.gov/9.

See Also

BesselI etc; the Hankel functions Hankel.

The CRAN package Rmpfr has Ai(x) for arbitrary precise "mpfr"-numbers x.

Examples

## The AiryA() := Ai() function -------------

curve(AiryA, -20, 100, n=1001)
curve(AiryA,  -1, 100, n=1011, log="y") -> Aix
curve(AiryA(x, expon.scaled=TRUE), -1, 50, n=1001)
## Numerically "proving" the 1st identity above :
z <- Aix$x; i <- z > 0; head(z <- z[i <- z > 0])
Aix <- Aix$y[i]; zeta <- 2/3*z*sqrt(z)
stopifnot(all.equal(Aix, 1/pi * sqrt(z/3)* BesselK(zeta, nu = 1/3),
                    tol = 4e-15)) # 64b Lnx: 7.9e-16;  32b Win: 1.8e-15

## This gives many warnings (248 on nb-mm4, F24) about lost accuracy, but on Windows takes ~ 4 sec:
curve(AiryA(x, expon.scaled=TRUE),  1, 10000, n=1001, log="xy")

## The AiryB() := Bi() function -------------

curve(AiryB, -20, 2, n=1001); abline(h=0,v=0, col="gray",lty=2)
curve(AiryB, -1, 20, n=1001, log = "y") # exponential growth (x > 0)

curve(AiryB(x,expon.scaled=TRUE), -1, 20,    n=1001)
curve(AiryB(x,expon.scaled=TRUE),  1, 10000, n=1001, log="x")