Reblogged from The Endeavour by J. D. Cook.

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 *x*with 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:

and

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 `Ai2`

and `Bi2`

below uses `np.where`

instead of `if... else`

so that it can operate on NumPy vectors all at once. You can plot `Ai`

and `Ai2`

and see that the two curves lie on top of each other. The same holds for `Bi`

and `Bi2`

.

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 `Ai2`

and `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.

###### Related articles

- Relating Airy and Bessel functions (johndcook.com)

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