<p> Γ(x) [15 D] Alvaro </p>

d6e902ca-4707-49b1-bcf1-2c6a02cd21ba 10 4 5 decimal

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

Γ.κ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh 1 - 2 ζ * + 2 / : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ.κ 1 el Γ.κ κ 1 + el z κ + / κ 114 sum + (* zp 2^* e zg -^* eval 21 line :

Valid for Re(ζ)>0

acc 100 x Γ x 1 e / e int 2.7196687

considered sanity
Gamma(x) does NOT
exist exactly ! 0.5 - Γ 0.0001 Γ 2 π * Γ 12 Γ 41 mat 3.54490770181108 - 9999.42288323158 195.936115724443 39916799.9999992 41 mat

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

n 1121 mat :

n ρ 2

d ρ 1

d 211 mat

K 50 :

This formula is "literature style". Mathcad
executes correct, Smath does not ... Why ?
Symbolic expansion is NOT correct. 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 :

x 12 / :

xx 12 / - :

n d x hypergeom 1 x - ln - x / n d xx hypergeom 1 xx - ln - xx / 41 mat 1.15072828980712 1.38629436111989 1.15072828980712 0.810930216216328 41 mat

correct expansion, will process  Barnes. 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 / 1 k + Γ 2^x k^2 k + Γ k! * / * ≡

x c 11 - x ≤ (x 1 ≤ (& cases :

n d x hypergeom 1 k + Γ (2^x k^2 k + Γ k! (* / * k 0 K sum :

n d x hypergeom x c *

n d x hypergeom 1 x - ln - x / n d xx hypergeom 1 xx - ln - xx / 41 mat 1.38629436111989 1.38629436111989 0.810930216216326 0.810930216216328 41 mat