H. T. Davis Γa(x)

492fb7d6-bfeb-4400-966d-350f1f35f678 68 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.

]]> https://keisan.casio.com/exec/system/1180573447

3 appVersion 0.98.6179 ≡ timing 9 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

a x γ a Γ a x Γ - ≡

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 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 :

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^/ * ≡

ε 1012 -^: a x γ t a 1 -^t - exp * t ε x int : a x Γ t a 1 -^t - exp * t x 100 int : 31 line

Γ x 0.01 0.5 0.02 range : i 1 x rows range g i el 0.5 x i el Γ eval : for x g augment 31 line :

γ x ε 1 0.05 range : i 1 x rows range g i el 0.5 x i el γ eval : for x g augment 31 line :

Γ Γ :

γ γ :

Γ 0 1.77245 + 40 red 1 5 mat 2 1 sys γ

1.709
1.573
1.16 0.5 0.001 Γ 0.5 0.01 Γ 0.5 0.1 Γ 31 mat 2.012 1.599 1.161 31 mat

a x γ a Γ a x Γ - ≡

Int(f,a,b,n) ... Simpson/Romberg Kernel [Jean]

Basic Romberg-Simpson ... integration acc from options ... NUMERIC f a b Romberg w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline 
Integration accuracy from user command line 'n' ... NUMERIC f a b n Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : ... α 0 : β 1 : h β α - n / : 15 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

x f x 0.5 1 -^x - exp * :

sanity [0.5] 1.770 f 106 -^10 Romberg 1.770

x f x 0.25 1 -^x - exp * :

sanity [0.25] 3.499 f 106 -^10 Romberg 3.478

∫Adaptive(f,a,b,N,ε) Carlos

f a b N ε ∫Adaptive sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 41 line eval :

eval(,) gains 1 min
.....
N 100 plenty
'ε' integration accuracy
.....
'a' infinite limit
'b' user end canvas plot x f x 0.5 1 -^x - exp * eval : N 100 : ε 0.001 : a 1015 -^: b e : 51 sys

x c 1 a x ≤ (x 2 ≤ (& cases :

x Canvas 1.77425 f a x N ε ∫Adaptive - :

t0 1 time :

user point Γ(0.5,x) 1 Canvas 0.280

x f x Canvas x c * 2 1 sys $1 time t0 - min 2.4$ $0.75 0.5 0.25 0.125 41 mat 1.22542 1.77425 3.62561 7.53394 41 mat ≡$