#4676 Some math functions which are not documented

[Robert Dodier]

There are some math functions which aren't documented.

• make_elliptic_e, make_elliptic_f, jacobi_am, kelliptic
• hypergeometric_regularized, hypergeometric_u
• whittaker_m, whittaker_w
• inverse_erf, inverse_erfc
• double_factorial, eulerpoly, cubrt

whittaker_m and whittaker_w are the same as %m and %w, respectively. There is documentation under the latter names.

Looks like there was an attempt some time ago to rename %m and %w but it was incomplete. Assuming we go ahead with that, then relabel the existing documentation to the new names.

I seem to recall double_factorial is not the same as the !! operator -- they handle different special cases, if I recall correctly. Probably should merge double_factorial into the simplification rules for !! and then nuke double_factorial.

[RoyalX]

double_factorial is a more powerful function than !!.

Unlike !!, double_factorial behaves like Factorial2 / !! from Mathematica - that is, it can also accept negative numbers as an argument, with the exception of negative even integers.

double_factorial also handles complex numbers correctly.

I think nothing needs to be removed. Both double_factorial and the !! operator should be kept. This is exactly the approach taken in Mathematica, for instance - not everyone prefers to use unary operators.

Therefore, double_factorial simply needs to be documented, while the !! operator should be brought into conformance with double_factorial, not the other way around.

[RoyalX]

There is a minor issue with the inverse_erf and inverse_erfc functions. The problem is that Maxima is unable to simplify inverse_erf(erf(x)) and inverse_erfc(erfc(x)) symbolically.

Additionally, some problems arise when attempting numerical evaluation.

First, there is significant numerical error: inverse_erf(erf(5)),numer evaluates to 5.000004776115367.

Second, inverse_erf(erf(6)),numer and inverse_erfc(erfc(28)),numer cannot be evaluated at all, since due to floating-point error the inverse functions receive 1 and 0 , respectively, as their arguments, at which point these functions are undefined.

Accordingly, a way should be found to allow Maxima to simplify these expressions symbolically.