Negative Zeros of the Airy Functions

Calculates the negative zeros of the Airy functions Ai(Z) or Bi(Z) or the negative zeros of the derivative Airy functions Ai'(Z)=0 or Bi.

NOTE: Here, Airy Function Ai(x) and Bi(x) that are linear independent solutions to the differential equation d^2y/dx^2 -x*y = 0 (the Stoke's equation) and NOT the Airy Disk Function that models the diffraction of waves through a circular aperture and NOT the Airy Stress Functions of material sciences.

This function calculates the negative zeros of the Airy functions Ai(Z) or Bi(Z) or the negative zeros of the derivative Airy functions Ai'(Z)=0 or Bi'(Z)=0. Ai(Z), Ai'(Z) have zeros on the negative real axis only. Bi(Z) and Bi'(Z) have zeros on the negative real axis and in the sector [pi/3< |Z| < pi/2]. This function returns the negative zeros.

inputs
K - type of Airy function
K=0 zeros of Airy function, Ai(Z), which is the same as airy(Z).
K=1 zeros of 1st derivative of Airy function, Ai'(Z).
K=2 zeros of Airy function of the second kind,Bi(Z)
K=3 zeros of 1st derivative of Airy function of the second kind, Bi'(Z)

N - index of the zero (vectorized)

The first 20 values are pretabulated. Values for indices larger than 20 are calculated using equations 10.4.94) etc. on page 450 of Abramowitz and Stegun, Handbook of Functions (this page is included in the download). Pretabulated values are given on page 478.

This is a PI-Day celebration project.