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Applications overlord . . . Γ(x)

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Gamma Function x Γ k 1 n 15 : (1 - (range c k el 1 - (k 1 -^k 1 - (! / k - n + (k 12 / -^* k - n + exp * : for z x 1 - (: ε 10 n -^: z n + (z 12 / +^z - n - exp * 2 π * sqrt c k el z k + / (ε + k 1 n 1 - sum + (* 41 line :

Lower Incomplete Gamma Function a x γ x a^x - exp * x k^a j + (j 0 k product / k 050 sum * :

Upper Incomplete Gamma Function a x Γ a Γ a x γ - :

Example_1: Multistage XFR

T 1 :

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

Example_2: Fitting a data set.

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 :

Fitting a data set Γ(a,x) ... data Mathsoft

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 143.017 : b 1.7666 : n 2.486 : 14 mat

x f yo a b x Γ n^* + :

Ω u u f vectorize augment :

u ε 3 0.05 range :

pts Ω 2 1 sys

yo 10.028 : a 143.01681 : b 1.766652 : n 2.48644 : 14 mat