Unclear docstring for airyx

[giordano]

The docstring of airyx function reads:

scaled k-th derivative of the Airy function, return \operatorname{Ai}(x) e^{\frac{2}{3} x \sqrt{x}} for k == 0 || k == 1\ , and \operatorname{Ai}(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|} for k == 2 || k == 3.

This is at best unclear, if not misleading. airyx(2, x) isn't the scaled second derivative of Airy Ai function, but rather the scaled Airy Bi function, like airyx(3, x) isn't the scaled third derivative of Airy Ai.

In addition, how is the scaling factor computed? For negative real values of x it should be complex according to the docstring (at least in the case k == 0 || k ==1), but airyx(0, -1.23) is real. GSL uses 1 as scaling factor for negative real x, but this isn't the case for Julia's airyx, because airyx(0, -1.23) != airy(0, -1.23).

[tkelman]

cc @simonbyrne or @stevengj might be the most knowledgeable here

[giordano]

Actually also the docstring of airy(k, x) is wrong: airy(2, x) isn't the first and second derivative of airy(1, x) and airy(0, x), respectively, it's just the airybi function. The same applies to airy(3, x)

[stevengj]

Yeah that looks right at first glance in the source code. Can we rename these horrible functions? It looks like the Fortran interface was copied a little too literally (or is it Matlab's fault?)—I'd rather just define eight functions in Julia (for Ai, Ai', Bi, Bi', and the scaled versions) rather than pass these obscure integer arguments.

The scaling is defined by the kode=2documentation in zairy.f and zbiry.f, I believe.

[tkelman]

These should be moved to a special functions package where they can be named more idiomatically.

[giordano]

These should be moved to a special functions package where they can be named more idiomatically.

Actually I like about Julia that you don't have to install and load an external package in order to use common special functions.

[tkelman]

Sorry, that's going to change, there's zero technical reason for them to be here and tied to the development schedule of the base language.