Airy function

Airy functions

Airy functions as per the Gnu Scientific Library, reference manual section 7.4 and AMS-55, section 10.4. These functions are declared in header file gsl_sf_airy.h

airy_Ai(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai(x, mode=0, give=FALSE, strict=TRUE)
airy_Bi_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai_deriv(x, mode=0, give=FALSE, strict=TRUE)
airy_Bi_deriv(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai_deriv_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_Bi_deriv_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_zero_Ai(n, give=FALSE, strict=TRUE)
airy_zero_Bi(n, give=FALSE, strict=TRUE)
airy_zero_Ai_deriv(n, give=FALSE, strict=TRUE)
airy_zero_Bi_deriv(n, give=FALSE, strict=TRUE)

Arguments

• x: input: real values

• n: input: integer values

• give: Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number

• mode: input: mode. For GSL_PREC_DOUBLE, GSL_PREC_SINGLE, GSL_PREC_APPROX use 0,1,2

  respectively

• strict: Boolean, with TRUE meaning to return NaN

  if status is an error

Details

The zero functions return a status of GSL_EDOM and a value of NA for $n <= 0$.

An example is given in the package vignette.

References

https://www.gnu.org/software/gsl/

Author(s)

Robin K. S. Hankin

Examples

x <- seq(from=0,to=1,by=0.01)

f <- function(x){
cbind(x=x, Ai= airy_Ai(x), Aidash= airy_Ai_deriv(x),
Bi=airy_Ai(x),Bidash=airy_Bi_deriv(x))
}

f(x)  #table 10.11, p475
f(-x) #table 10.11, p476

x <- 1:10  #table 10.13, p478
cbind(x,
 airy_zero_Ai(x), airy_Ai_deriv(airy_zero_Ai(x)),
 airy_zero_Ai_deriv(x), airy_Ai(airy_zero_Ai_deriv(x)),
 airy_zero_Bi(x), airy_Bi_deriv(airy_zero_Bi(x)),

 airy_zero_Bi_deriv(x), airy_Bi(airy_zero_Bi_deriv(x))
 )

# Verify 10.4.4 and 10.4.5, p446:
3^(-2/3)/gamma(2/3)   - airy_Ai(0)
3^(-1/3) / gamma(1/3) + airy_Ai_deriv(0) 
3^(-1/6) / gamma(2/3) - airy_Bi(0)
3^(1/6) / gamma(1/3)  - airy_Bi_deriv(0)
#  All should be small