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

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

Arguments

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

## 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")

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(Bessel)
Loading required package: Rmpfr
Loading required package: gmp

Attaching package: 'gmp'

The following objects are masked from 'package:base':

    %*%, apply, crossprod, matrix, tcrossprod

C code of R package 'Rmpfr': GMP using 64 bits per limb


Attaching package: 'Rmpfr'

The following objects are masked from 'package:stats':

    dbinom, dnorm, dpois, pnorm

The following objects are masked from 'package:base':

    cbind, pmax, pmin, rbind

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/Bessel/Airy.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Airy
> ### Title: Airy Functions (and Their First Derivative)
> ### Aliases: Airy AiryA AiryB
> ### Keywords: math
> 
> ### ** Examples
> 
> ## 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")
There were 50 or more warnings (use warnings() to see the first 50)
> 
> 
> ## 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")
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>