Γ(x) [15 D] Alvaro Original

37326b3c-4be6-4bca-af5b-b15baf8af692 9 3 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 z 0.5 + : 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

Γ(x) [15 D] Alvaro Same otherwise

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

Valid for Re(ζ)>0

x Γ

sanity 15 Digits 0.125 Γ 7.53394159879757

Smath 6179 0.125 Gamma 7.53395331627959

x Gamma vectorize x 1 π int 2.611

x Γ x 1 π int 2.611

============================================================================

Inifinitesimal Analysis . . . to rescue d/dx(G(x)

]]>

constrain to validate PDF hypergeometric x Γ x Γ 0 x ≤ (* :

Plots quicker, NO need constrain x Γ x Gamma :

Hypergeometric PDF ...

N n p x hypergeom N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 60 :

n 10 :

p 5 :

x f N n p x hypergeom eval :

h 0.01 :

1rst derivative x D1 x 2 h * - f x 2 h * + f - 4 h * / - :

2nd derivative x D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

Continuous scalar PDF Hypergeometric. Initialise Golden ratio by inspection.

x f x D1 x D2 3 1 sys

Psi(x) Digamma for complex 'z' x Γ x diff x Psi x Γ * ≡

2 i * Psi maple 0.715 1.821 i * +

1. solve fails infinitesimal D1(x)
2. dichotomy fails as well.
3. Golden ratio succeeds. x D1 0 ≡ x 0.1 1 solve #

x f x 05 int 0.7982

Does not
integrate
rescue FD x f x 1 - 0 int #

0.25 - f 0.361

Golden ratio

The hyperfast Golden-Ratio search [min/max]

x f N n p x hypergeom : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ max in range ≡ y1 y2 ≤ min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 61 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

Ex f 0.25 0.5 y1 y2 ≥ 1012 -^Extrema 1 el :

Ex Ex :

Ex Ex f 12 mat 0.386297 0.504888 12 mat

incertitude sur N mesures ayant σ

N 12 :

σ 0.75 :

2 N 1 - / sqrt N 2 / Γ N 1 - 2 / Γ / * σ * 0.7332

========================================

Ucertainty on vector of μ & σ

vy 25.527 25.51 25.512 25.511 25.511 25.508 25.487 25.501 81 mat :

vy incertitude n vy length : u vy StdDev n sqrt / n 1 - 2 / Γ n 1 - sqrt * n 2 / Γ 2 sqrt * / * : 21 line :

vy incertitude 0.00413

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Since the sampling is done without replacement, the probability of finding a given number of defectives is not well modeled by the Poisson distribution when the sample size is an appreciable fraction of the lot size. In this case, the Hypergeometric distribution provides a good model for the sampling process. The application below has been modified to model sampling from small lots using the hypergeometric distribution

Lot size N 40 :

number of defectives D 0 N range :

sample size n 8 :

acceptance criterium c 2 :

The statistical Hypergeometric

n k cb n! k! (n k - (! * / n k ≥ 0 cases :

N n D u H D u cb N D - n u - cb * N n cb / :

D PA 1 D c ≤ c i - (N D - (≤ (N n D i H * (i 0 c sum cases :

Lot Acceptance Probability LAP D N / D PA vectorize augment :

LAP 01 0.025 1 0.05 1 0.075 0.994 0.1 0.98 0.125 0.954 0.15 0.918 0.175 0.872 0.2 0.817 0.225 0.755 0.25 0.688 112 mat

Probability of acceptance vs defect rate

LAP LAP o 5 red plot 2 1 sys