<p> H. T. Davis Gamma(x) </p>

4d55ae19-999f-4c31-b424-66b860813f02 22 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Abramowitz & Stegun, p. 256 c 1 0.5772156649015329 0.6558780715202538 - 0.0420026350340952 - 0.1665386113822915 0.0421977345555443 - 0.0096219715278770 - 0.0072189432466630 0.0011651675918591 - 0.0002152416741149 - 0.0001280502823882 0.0000201348547807 - 0.0000012504934821 - 0.0000011330272320 0.0000002056338417 - 0.0000000061160950 0.0000000050020075 0.0000000011812746 - 0.0000000001043427 0.0000000000077823 0.0000000000036968 - 0.0000000000005100 0.0000000000000206 - 0.0000000000000054 - 0.0000000000000014 0.0000000000000001 261 mat :

1 z Γ / c k el z k^* k 1 ∞ sum ≡

z Γpoly c k el z k^* k 126 sum :

z Γa 1 z Γpoly / :

x c 10 x ≤ (x 1 ≤ (& cases :

x Γpoly x c *

Gamma(x)

x Γ x Γa 0 x ≤ (x 1 ≤ (& x 1 - (x 1 - Γa * 1 x ≤ (x 2 ≤ (& x 2 - (x 1 - (* x 2 - Γa * 2 x ≤ (x 3 ≤ (& x 3 - (x 2 - (* x 1 - (* x 3 - Γa * 3 x ≤ (x 4 ≤ (& x 4 - (x 3 - (* x 2 - (* x 1 - (* x 4 - Γa * 4 x ≤ (x 5 ≤ (& x 5 - (x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 5 - Γa * 5 x ≤ (x 6 ≤ (& x 6 - (x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 6 - Γa * 6 x ≤ (x 7 ≤ (& x 7 - (x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 7 - Γa * 7 x ≤ (x 8 ≤ (& x 8 - (x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 8 - Γa * 8 x ≤ (x 9 ≤ (& x 9 - (x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 9 - Γa * 9 x ≤ (x 10 ≤ (& hello ! cases :

0.5 Γ 1.5 Γ 2.5 Γ 3.5 Γ 4.5 Γ 5.5 Γ 6.5 Γ 7.5 Γ 8.5 Γ 10.5 Γ 101 mat 1.77245385090552 0.886226925452759 1.32934038817914 3.32335097044785 11.6317283965675 52.3427777845536 287.885277815045 1871.25430579779 14034.4072934834 hello ! 101 mat

3 Γ 2

x c 10 x ≤ (x 5 ≤ (& cases :

min Γ(x 1.461632144968362 0.885603194410889 12 mat

π Γ 2.28803779534003

Γ(x) 15D ... sanity checked vs Mathcad 11 Abramowitz & Stegun Table 6.1 is only 10 D

x Γ x c * 1 1 o 5 red 1 5 mat 2 1 o 5 red 1 5 mat 3 2 o 5 red 1 5 mat 4 6 o 5 red 1 5 mat 5 24 o 5 red 1 5 mat 1.4 0.885 . 14 black 1 5 mat 7 1 sys

x c 10 x ≤ (x 1 ≤ (& cases :

x Γpoly x c *

x c 10 x ≤ (x 5 ≤ (& cases :

Does not compute Γ(integer)... /0 x Γ i 1 x Floor range r i el i : for x r rows - Γa x r n el - (n 1 r rows product * 21 line :

x Γ x c * 8 2 1 sys

π Γ 0.5 Γ π sqrt 31 mat 2.28803779534003 1.77245385090552 1.77245385090551 31 mat

x f 8 :

x Gamma vectorize x f - 0 ≡ x 05 solve x Γ vectorize x f - 0 ≡ x 05 solve 21 mat 0.118 4.2235 21 mat 0.118 4.2235 21 mat 21 mat

The 15 D Γ(x) above is not long enough to enable the Barnes hypergeometric.

n & d are supplied in vector form

m 1121 mat :

v ρ v rows : d 21 identity * : 21 line

x 12 / :

xx 12 / - :

K 8 :

m d x hypergeom m i el k + Γ x k^* m i el Γ / i 1 m ρ product d j el k + Γ d j el Γ / k! (* j 1 d ρ product / k 0 K sum 11 line :

m d x hypergeom 1 x - ln - x / m d xx hypergeom 1 xx - ln - xx / 41 mat #