H. T. Davis Γa(x)

492fb7d6-bfeb-4400-966d-350f1f35f678 59 15 12 decimal

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

ε 1013 -^: a x γ t a 1 -^t - exp * t ε x int eval : a x Γ t a 1 -^t - exp * t x 100 int eval : 31 line

Numerical . . .

. . . Γ(a,x) will compute and plot only '1 ≤ a' .

ltypical upper limit [100] acc 1000 => app/demo below.

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3 appVersion 0.98.6179 ≡ timing 5 min acc 1000 ≡ 21 line

ε 1012 -^:

a 1 : x 0.5 : 12 mat

a x γ a x Γ 21 mat 0.3935 0.6065 21 mat

x ε 4 0.1 range :

γax x a x γ vectorize augment : Γax x a x Γ vectorize augment : 21 sys

γax Γax 2 1 sys

x sanity 112 / x 2^Γ π sqrt / - :

1.23456789 sanity 1.23456789 erf 21 mat 0.91917907 0.91917908 21 mat

H. T. Davis Γa(x)

Abramowitz & Stegun, p. 256 c 1 0.5772156649015329 0.6558780715202538 - 0.0420026350340952 - 0.1665386113822915 0.0421977345555443 - 0.0096219715278770 - 0.0072189432466630 0.0011651675918591 - 0.0002152416741149 - 0.0001280502823882 0.0000201348547807 - 0.0000012504934821 - 0.0000011330272320 0.0000002056338417 - 0.0000000061160950 0.0000000050020075 0.0000000011812746 - 0.0000000001043427 0.0000000000077823 0.0000000000036968 - 0.0000000000005100 0.0000000000000206 - 0.0000000000000054 - 0.0000000000000014 0.0000000000000001 261 mat :

1 z Γ / c k el z k^* k 1 ∞ sum ≡

z Γpoly c k el z k^* k 126 sum eval :

z Γa 1 z Γpoly / :

x c 10 x ≤ (x 1 ≤ (& cases :

x Γa x c * 1 1 + 15 red 1 5 mat 2 1 sys if/otherwise ... (x ≤ 0)&(x ≤ 15) suite IterativeProduct

Gamma(x)

constant 15 D accuracy @ integers ... can be made iterated product(s) x Γ x Γa x 0 ≤ x Γa 0 x ≤ (x 1 ≤ (& x 1 - (x 1 - Γa * 1 x ≤ (x 2 ≤ (& x 2 - (x 1 - (* x 2 - Γa * 2 x ≤ (x 3 ≤ (& x 3 - (x 2 - (* x 1 - (* x 3 - Γa * 3 x ≤ (x 4 ≤ (& x 4 - (x 3 - (* x 2 - (* x 1 - (* x 4 - Γa * 4 x ≤ (x 5 ≤ (& x 5 - (x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 5 - Γa * 5 x ≤ (x 6 ≤ (& x 6 - (x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 6 - Γa * 6 x ≤ (x 7 ≤ (& x 7 - (x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 7 - Γa * 7 x ≤ (x 8 ≤ (& x 8 - (x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 8 - Γa * 8 x ≤ (x 9 ≤ (& x 9 - (x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 9 - Γa * 9 x ≤ (x 10 ≤ (& x 10 - (x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 10 - Γa * 10 x ≤ (x 11 ≤ (& x 11 - (x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 11 - Γa * 11 x ≤ (x 12 ≤ (& x 12 - (x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 12 - Γa * 12 x ≤ (x 13 ≤ (& x 13 - (x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 13 - Γa * 13 x ≤ (x 14 ≤ (& x 14 - (x 13 - (* x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 14 - Γa * 14 x ≤ (x 15 ≤ (& x 15 - (x 14 - (* x 13 - (* x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 15 - Γa * cases :

Γ(x)=> usable as is up to x = 15 Higher numerical: use iterated product

x Γ

1/1 Mathcad sanity up to 15 14.987654321 Γ 84347399235.0809

x Γ ext'd iter'd product n x trunc : x j - (j 1 n product x n - Γa * eval 31 line :

Numerical Γ(x) . . .

Sanity vs Mathcad => . . . not exhausted

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29.6 Γ 2.2897321973268 1030^*

Doctor 640 works for  a =  1.767.
For other 'a'  Doctor may be else ?
Depends upon "acc"... here 2000. b x Γ t b 1 -^t - exp * t x 100 int eval :

0.923
0.9136
0.80126 1.767 0 Γ 1.767 0.1 Γ 1.767 0.5 Γ 31 mat 0.92291 0.91396 0.80126 31 mat

a x Γ t a 1 -^t - exp * t x 100 int eval :

x 05 0.1 range : Γax x 3 x Γ vectorize augment : 21 sys

Γax

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 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 :

b x Γ t b 1 -^t - exp * t x 100 int eval :

pts XY o 5 black plot :

1.767 0 Γ 0.9229

yo 10.18722 : a 143.01681 : b 1.766652 : n 2.48644 : 14 mat

x f yo a b x Γ n^* + :

Ω u u f vectorize augment :

u 03 0.05 range :

pts Ω 2 1 sys

1000 ≤ acc a x Γ t a 1 -^t - exp * t x 100 int eval :

x 05 0.1 range :

Γx1 x 1 x Γ vectorize augment : Γx2 x 2 x Γ vectorize augment : Γx3 x 3 x Γ vectorize augment : Γx35 x 3.5 x Γ vectorize augment : Γx4 x 4 x Γ vectorize augment : 51 sys

Γx1 Γx2 Γx3 Γx35 Γx4 5 1 sys

sanity check Mathcad 15 Γ 25 Γ 35 Γ 3.5 0 Γ 3.5 5 Γ 45 Γ 61 mat 0.006738 0.040428 0.249304 3.32335 0.626696 1.590155 61 mat

n 9 :

T 1 :

XFR of 'n' multistage system of
=  Unit Time constant 'T' [1] t nth 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum 11 line * - :

Reaction rate of the nth stage n t Laplace t n 1 - (^n 1 - (! / t - T / exp T n^/ * :

Numeric Local_definition 21 sys

t nth 1 t Laplace 3 t Laplace 5 t Laplace 7 t Laplace 9 t Laplace 6 1 sys

XFR . . . Transfer Function

Is the response of a dynamic system to unit step.

In Laplace algebra, unit step = 1/s.

XFR the graphical response w/o oscillation over

the time domain ' t '.

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

=> invlaplace 1 t - T / exp 1 k! / t T / (k^* (k 1 n sum * -

individual stage 
reaction rate T n s d 1 T s * 1 + (n^/ :

inverselaplace t n 1 - (^n 1 - (! / t - T / exp T n^/ * ≡