<content> Gamma(x) Lanczos overlord </content>

8177e8cb-605a-49b3-8741-c79e6bd91c3b 7 overlord 4 5 false false 0 decimal

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

Gamma Function #_p 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 -^* 72 mat : #a Γ #S ∞ : #a 0 ≤ (#a #a trunc ≡ (& #S #a 1 - (! : #a #a trunc ≡ #a.T #a - #a 0.5 < #a 1 - cases : #fct 1.5 #a 165 - < (#a 165 > (| 0 cases : 21 line cases #img 0 : 21 line #a Im 0 ≡ #a.T #a 1 - : #fct 0 : #img 1 : 31 line cases #S #S IsDefined #x 0.99999999999999709182 #_p #i el #a.T #i + / #i 114 sum + : #g 4.7421875 : #a.T #a.T 0.5 + : #S #a.T #g + e / (#a.T^e #g -^2 π * sqrt * #x * (* : #fct 0 ≡ #S #a.T #g + e / (#a.T #fct /^e #g -^2 π * sqrt * #x * (1 #fct /^* (#fct^: cases #S #img 1 ≡ π π #a * sin #S * / #a 0.5 < (#S cases cases 51 line cases 21 line : 21 line

Upper Incomplete Gamma Function #a #x Γ #a Γ #a #x γ - 11 line :

Lower Incomplete Gamma Function ##a ##x γ #SS 0 : #k 070 range #TT 1 : #j 0 #k range #TT #TT ##a #j + (* : for #SS #SS ##x #k^#TT / + : 31 line for #SS ##x ##a^* e ##x -^* 31 line :

<content> X(Φ,α,β,ε) ..... RootSecant algo </content>

"RootSecant" is viable  => guess wisely tial/error Φ α β ε X Root Secant u 1 el ε : u 2 el β β Φ α β - (* α Φ β Φ - / - : i# 2 : u i# 1 + el u i# el u i# el Φ u i# 1 - el u i# el - (* u i# 1 - el Φ u i# el Φ - / - : u i# el u i# 1 - el - abs ε ≥ u i# 1 + el u i# el u i# el Φ u i# 1 - el u i# el - (* u i# 1 - el Φ u i# el Φ - / - : i# i# 1 + : 21 line while u 71 line eval :

z 2.5 i * :

4 appVersion 0.99.7822.147 ≡ Applications 8 s ≡ 21 mat

z Γ 0.0161290540478 0.0267491089459 i * -

a 0.5 :

x Φ a x γ a x Γ - eval :

α 0.2 : β 0.5 : ε 1015 -^: 13 mat

sol Φ α β ε X :

Explode the nested 'sol' R sol sol rows row 1 el :

sol 11015 -^* 0.238739753660938 0.233859672474681 31 mat

R a R Γ 12 mat 0.234 0.876 12 mat

a x Γ a x γ R a R Γ + 20 black 1 5 mat 3 1 sys

Example_1: Multistage XFR

t0 1 time :

System TimeConstant T 1 :

Laplace UnitStep s 1 :

x λ 10 x ≤ (x 20 ≤ (& cases :

time 7 :

Laplace XFR multistage system'n' stages, 'T' time constant n t f n 1 + Γ n 1 + t T / Γ - s n 1 + Γ * / :

3 x f x λ * 6 x f x λ * 9 x f x λ * time 1.25 time 0.25 - 2 2 mat 6 1 sys

3 time f 6 time f 9 time f 31 mat 0.92 0.55 0.17 31 mat

0.92 0.55 0.17 31 mat

XY 0 127.42 0.1 123.469 0.2 118.008 0.3 111.187 0.4 102.811 0.5 93.554 0.6 84.168 0.7 74.818 0.8 66.373 0.9 58.587 1 51.678 1.1 45.739 1.2 40.675 1.3 36.404 1.4 32.74 1.5 29.581 1.6 26.848 1.7 24.472 1.8 22.395 1.9 20.573 2 18.97 2.1 17.555 2.2 16.3 2.3 15.184 2.4 14.188 2.5 13.295 2.6 12.492 2.7 11.768 2.8 11.12 2.9 10.547 3 10.028 312 mat :

Example_2: Fitting Mathsoft data set.

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 :

pts XY o 5 black plot :

ε 1015 -^:

yo 10.028 : a 143 : b 1.77 : η 2.5 : 14 mat

x f yo a b x Γ η^* + :

Ω u ε 3 0.05 range : i 1 u rows range vy i el u i el f eval : for u vy augment 31 line :

X ε 3 0.05 range : i 1 X rows range Y i el X i el f eval : for x S X Y x cinterp : 31 line

... 'G' ▼  ordinate solve abscissa G 64 :

U x S G - 0 ≡ x ε 3 solve : 0.836

plot 0 G U G U 16 - 32 mat Ω pts 0.175 - G 7 + G 12 blue 15 mat U 0 U 12 blue 15 mat 51 sys :

plot

G U 12 mat 64 0.836 12 mat

Example_3: Hypergeometric statistical function.

Hypergeom ...

no need for this x Γ x Gamma :

this neither x Γ x Lanczos :

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 39 :

n 12 :

p 20 :

x hypergeom N n p x hypergeom :

Continuous PDF Hypergeometric

0 x ≤ x hypergeom outside lower bound if

In probability theory and statistics,
the hypergeometric distribution is a discrete
probability distribution that describes the
probability of 'k' successes in 'n' draws,
without replacement, from a finite population
of size 'N' that contains exactly 'K' successes,
wherein each draw is either a success or a failure.
In contrast, the binomial distribution describes
the probability of 'k' successes in 'n' draws
with replacement.

1 time t0 - 8 s *

RemToDo ▲ code recursive