<p> Γ(x) [15 D] Alvaro "The very best known" </p>

55e2effc-c1d8-443f-a42a-788931d167bc 42 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

Gauss Hypergeometric Function

x Γ 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 $x c 1 0.875 - x ≤ (x 1 ≤ (& cases :$ $Smath does not support Integral representation
The upper limit of Alvaro Γ(x) ∞ ... e307
Smath 6179 upper Σ [98]... above max allowed. a b c x fhyper c Γ a Γ b Γ * / a n + Γ b n + Γ * c n + Γ / x n^n! / (* n 098 sum * :$ iVBORw0KGgoAAAANSUhEUgAAABUAAABBCAYAAAA+CO93AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACzSURBVFhH7Y5tCsQgDAW9/0G9hstbdEndl4/a2h8lAwOpxqGlGdRa+3QOMyopJbya0SAZ7ZNPRmNktE8+L4siuC0aDbtbMrgtGgmbGywIPZai0EK9ZSGpxXIUatAbFmBqXIpCxt8pe2jJuByFM4cT9iDizC1RKPl9scUzSm6LwsF3YgsrDo7/PcEeQo+M6rAg9MioDgtCj4zqsCD0eD4qicQGGY2R0T75ZDRGRvvksyHa2gf6ouW+V7sCJQAAAABJRU5ErkJggg==

Mathcad fhyper(a,b,c,x) is NOT based on the Integral representatio

a 0.5 : b 0.5 : c 1 : 13 mat

a b c x fhyper x c * 1 5 1 1 - 2 2 mat a b 0.75 x fhyper x c * 3 1 sys

0.834627
1.033658
5.280157 a b c 1 - fhyper a b c 0.125 fhyper a b c 0.999999 fhyper 31 mat 0.836238 1.033658 2.528114 31 mat

https://keisan.casio.com/exec/system/1349143084

Pochhammer original symbol x n x.n x k + (k 0 n 1 - product :

x y XY x y / - :

x 1 x.n x 2 x.n x 3 x.n x 4 x.n x 5 x.n 5 1 sys

0.5 8 x.n 7918.0664

https://keisan.casio.com/exec/system/1180573462

x 5 x.n x 1 x + (* 2 x + (* 3 x + (* 4 x + (*

x 1 x.n x 2 x.n x 3 x.n x 4 x.n x 5 x.n XY 6 1 sys

Kummer Confluent Hypergeometric Function

x c 10 x ≤ cases :

a b x mhyper a n + Γ b Γ * a Γ b n + Γ * / x n^n! / (* n 020 sum :

3.5 - 1 x mhyper x c * 4.5 - 1 x mhyper x c * 0.1 5.5 - 1 x mhyper * x c * 3 1 sys

sanity 23 0.5 mhyper 1.405115

https://keisan.casio.com/exec/system/1349143651 2.5 - 1 x mhyper x c * 1 1 sys