67ce8613-73d9-4015-9bf6-4fe952e360ab 14 4 5 decimal

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

https://en.wikipedia.org/wiki/Spouge%27s_approximation

https://deamentiaemundi.wordpress.com/2013/06/29/the-gamma-function-with-spouges-approximation/

z 1 + Γ z a + (z 12 / +^e z a + (-^* c.0 c.k z k + / k 1 a 1 - sum + z ε.a + (* ≡

c.0 2 π * sqrt :

c.k 1 - (k 1 -^k 1 - (! / k - a + (k 12 / -^* e k - a +^* ≡

ε a 12 / -^2 π * (a 12 / + (-^* ≤

if

z Re 0 > (a 2 > (&

ε 1018 -^:

approx (see https://deamentiaemundi.wordpress.com/2013/06/19/on-doing-it-yourself/) a ε ln 2 π * ln / - Ceil : 23

a 49 :

c.0 2 π * sqrt :

j 1 a 1 - (range c.k j el 1 - (j 1 -^j 1 - (! / a j - (j 12 / -^* e a j -^* : for

c.k 1.68147317110891 1010^* 1.2691503155033 1011^* - 4.33944942901303 1011^* 8.89368020269268 1011^* - 1.21847242586795 1012^* 1.17840347325935 1012^* - 8.28245531124626 1011^* 4.29211287893062 1011^* - 1.64698834727648 1011^* 4.66151492198911 1010^* - 9.61997256451544 109^* 1.41938255178104 109^* - 1.45414547081639 108^* 9.92302071943579 106^* - 4.25355756391909 105^* 10533.7105978434 - 133.242547953796 0.711834397402947 - 0.001172051640058 3.32394088582412 107 -^* - 4.50380167440434 1012 -^* 5.32047700221164 1020 -^* - 221 mat

z Γ res z a + (z 12 / +^e z a + (-^* : gamma(z+1) res res c.0 c.k k el z k + / k 1 a 1 - sum (+ (* : gamma(z) res z / 51 line :

x Γ x Gamma x Gamma x Γ - ( 10 0 ^ * 2 π * x / sqrt x e / ( x ^ *

32 / - Γ 2.36327180120727

12 / - Γ 3.54490770181088 -

0.5 Γ 1.7724538509053

π sqrt 1.77245385090552

1 Γ 0.999999999999759

1

n Γ n 1 - (! ≡

32 / Γ 0.886226925452577

2 Γ 0.99999999999863

1

52 / Γ 1.3293403881774

3 Γ 1.99999999999176

2

π Γ 2.28803779533067

72 / Γ 3.32335097042258

4 Γ 5.99999999990048

10 Γ 362879.999823191

9! 362880

0 Γ #

1 - Γ #

n/d

2 - Γ #

n/d

should be n/d for negative integers

3 - Γ #

n/d

∞ + Γ #

∞ +

∞ - Γ #

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

n 1121 mat :

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

d 211 mat

x 12 / :

xx 12 / - :

K 20 :

n d x hypergeom 1 x - ln - x / n d xx hypergeom 1 xx - ln - xx / 41 mat 1.15072828980736 1.38629436111989 1.15072828980736 0.810930216216328 41 mat

k kk : n i el k + Γ n i el Γ / x k^* i 1 n ρ product d j el k + Γ d j el Γ / k! (* j 1 d ρ product / 21 line #