>
>
> It's believed that such functions are best computed/approximated as continued
> fractions:
>
> -- http://people.math.sfu.ca/~cbm/aands/page_263.htm
>
> FriCAS can do this nicely ;-)
Well, this is only for positive x. In general there are several
methods. IIUC continued fractions in general converge quite
slowly (in restricted regions convergence is faster), so it
is hard to argue that this is best method. There are methods
based on asymptotic expansions, but they need apropriate error
bounds. There are published bounds for real arguments, but
for complex ones I am not aware of complete result.
One can treat incomplete Gamma as a generalised hypergeometric
function. Luke argued that one should use Chebyshev expansions
to compute such functions and published several books about
this subject. I had access to only one of his books and
this books gave details in several special cases, but:
- incomplete Gamma was not under consideration
- he did not gave general theory in this book
so I do not know how well this would work for incomplete Gamma.
We could try to implement generalised hypergeometric functions
via Taylor series + analytic continuation. This is not extremaly
hard, but takes nontrivial amount of code.
--
Waldek Hebisch
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