Relating Airy and Bessel functions
The Airy functions Ai(x) and Bi(x) are independent solutions to the differential equation
For negative x they act something like sin(x) and cos(x). For positive x they act something like exp(x) and exp(-x). This isn’t surprising if you look at the differential equation. If you replace xwith a negative constant, you sines and cosines, and if you replace it with a positive constant, you get positive and negative exponentials.
The Airy functions can be related to Bessel functions as follows:
Here J is a “Bessel function of the first kind” and I is a “modified Bessel function of the first kind.” Also
To verify the equations above, and to show how to compute these functions in Python, here’s some code.
The SciPy function
airy computes both functions, and their first derivatives, at once. I assume that’s because it doesn’t take much longer to compute all four functions than to compute one. The code for
Bi2 below uses
np.where instead of
if... else so that it can operate on NumPy vectors all at once. You can plot
Ai2 and see that the two curves lie on top of each other. The same holds for
from scipy.special import airy, jv, iv from numpy import sqrt, where def Ai(x): (ai, ai_prime, bi, bi_prime) = airy(x) return ai def Bi(x): (ai, ai_prime, bi, bi_prime) = airy(x) return bi def Ai2(x): third = 1.0/3.0 hatx = 2*third*(abs(x))**1.5 return where(x > 0, third*sqrt( x)*(iv(-third, hatx) - iv(third, hatx)), third*sqrt(-x)*(jv(-third, hatx) + jv(third, hatx))) def Bi2(x): third = 1.0/3.0 hatx = 2*third*(abs(x))**1.5 return where(x > 0, sqrt( x/3.0)*(iv(-third, hatx) + iv(third, hatx)), sqrt(-x/3.0)*(jv(-third, hatx) - jv(third, hatx)))
There is a problem with
Bi2: they return
nan at 0. A more careful implementation would avoid this problem, but that’s not necessary since these functions are only for illustration. In practice, you’d simply use
airy and it does the right thing at 0.
- Relating Airy and Bessel functions (johndcook.com)