(j3.2006) 13.7.64 GAMMA has incomplete mathematical definition
Van Snyder
Van.Snyder
Thu Oct 2 22:06:34 EDT 2008
The integral representation given for the Gamma function in the Result
Value paragraph of 13.7.64p5 is valid only for positive values of the
argument. For negative values, a Cauchy-Saalschuetz representation is
required. The first sentence of the paragraph should end after "gamma
function of X." A new sentence should begin just before the mathematical
definition "For $X > 0$, $\Gamma(X) = ..." Then another sentence should
be appended: "Otherwise, if $-k-1 < X < -k$ for some integer $k$,
$\Gamma(X) = \int_0^\infty t^{X-1} \left( \exp(-t) - \sum_{i=0}^k (-1)^i
\frac{t^i}{i!}\, \text{d} t \right)$."
See page 44 of "Special Functions" by Nico Temme.
I should have caught this when we introduced the GAMMA intrinsic. I
just didn't notice we had allowed it for all values other than negative
integer or zero, rather than only for positive values, of its argument.
Should we
1. Expand the definition in this way?
1a. By Malcolm as /Edit, or
1b. By a paper?
1b(i) Can I do the paper in LaTeX so you can read the formula, or
1b(ii) must I try to write the formula in ASCII, or
1b(iii) should I just leave the formula in LaTeX as above for you
to compile in your head?
or
2. Restrict the arguments of GAMMA and LOG_GAMMA to positive values?
Van
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