H. T. Davis Γa(x)

384cf127-1349-4426-a57d-c46f9348bd4b 102 overlord 4 5 decimal

&[DATE] &[TIME]

_p 676.5203681218851 1259.1392167224028 - 771.32342877765313 176.61502916214059 - 12.507343278686905 0.13857109526572012 - 9.9843695780195716 106 -^* 1.5056327351493116 107 -^* 42 mat :

Gamma Function a Γ a Re 0.5 < S π π a * sin 1 a - Γ * / : a.T a 1 - : x 0.99999999999980993 : i 1_p length range x x_p i el a.T i + / + : for t a.T 7.5 + : S 2 π * sqrt t a.T 0.5 + 2 t * /^e 12 / -^* (2 t *^* x * : 51 line if a a 01 round ≡ S a 1 - (! : S if 21 line :

Lower Incomplete Gamma Function a x γ x a^e x -^* x k^a j + (j 0 k product / k 050 sum * :

Upper Incomplete Gamma Function a x Γ a Γ a x γ - :

t.0 0 time :

165.123456789 Γ #

15 Γ #

0.928153 Γ #

0 time t.0 - ms 94

H. T. Davis Γa(x)

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 eval :

z Γa 1 z Γpoly / :

x c 10 x ≤ (x 1 ≤ (& cases :

x Γa x c * 1 1 + 15 red 1 5 mat 2 1 sys if/otherwise ... (x ≤ 0)&(x ≤ 15) suite IterativeProduct for convenience

Gamma(x)

constant 15 D accuracy @ integers ... can be made iterated product(s) x Γ x Γa x 0 ≤ 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 ≤ (& x 10 - (x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 10 - Γa * 10 x ≤ (x 11 ≤ (& x 11 - (x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 11 - Γa * 11 x ≤ (x 12 ≤ (& x 12 - (x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 12 - Γa * 12 x ≤ (x 13 ≤ (& x 13 - (x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 13 - Γa * 13 x ≤ (x 14 ≤ (& x 14 - (x 13 - (* x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 14 - Γa * 14 x ≤ (x 15 ≤ (& x 15 - (x 14 - (* x 13 - (* x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 15 - Γa * cases :

Γ(x)=> usable as is up to x = 15 Higher numerical: use iterated product

x Γ

1/1 Mathcad sanity up to 15 0.5 Γ 1.77245385090552

x Γb high acc 'x' fract'l n x trunc : x j - (j 1 n product x n - Γa * eval 31 line :

U 2 :

X x Γ U - 0 ≡ x 04 solve : 0.4429 321 mat

x Γ x Γa x 1 ≤ x Γb cases :

165.123456789 Γ 6.17233525292524 10293^*

x λ0 1 x 1 ≤ cases :

x λ1 11 x ≤ cases :

x Γ U 2 1 sys x Γb x λ1 * x Γa x λ0 * 1 4 1 4 - 2 2 mat 3 1 sys