Airy Functions (and Their First Derivative)

Description

Compute the Airy functions Ai or Bi or their first derivatives, d/dz Ai(z) and d/dz Bi(z).

Usage

1
2
AiryA(z, deriv = 0, expon.scaled = FALSE)
AiryB(z, deriv = 0, expon.scaled = FALSE)

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).

Details

By default, when expon.scaled is false, AiryA() computes the complex Airy function Ai(z) or its derivative d/dz Ai(z) on deriv=0 or deriv=1 respectively.
When expon.scaled is true, it returns exp(zta)*Ai(z) or exp(zta)* d/dz Ai(z), effectively removing the exponential decay in -pi/3 < arg(z) < pi/3 and the exponential growth in pi/3 < abs(arg(z)) < pi, where zta=(2/3)*z*sqrt(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 d/dz Bi(z) on deriv=0 or deriv=1 respectively.
When expon.scaled is true, it returns exp(-abs(Re(zta)))*Bi(z) or exp(-abs(Re(zta)))* dBi(z)/dz, to remove the exponential behavior in both the left and right half planes where, as above, zta=(2/3)*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 BesselI.

See Also

BesselI etc; the Hankel functions Hankel.

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
## The AiryA() := Ai() function

curve(AiryA, -20, 100, n=1001)
curve(AiryA,  -1, 100, n=1001, log="y")
curve(AiryA(x, expon.scaled=TRUE), -1, 50, n=1001)
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")