<content> H. T. Davis Γa(x) </content>

ea76b3eb-64ba-4bf5-a5f5-053596268f8f 34 4 5 false false 1 decimal

&[DATE] &[TIME]

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 16 - x ≤ (x 1 ≤ (& cases :

x Γa x c * 1 1 + 15 red 1 5 mat 2 1 sys <content> Lanczos Gamma Approximations </content>

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 ∞ a 0 ≤ (a a trunc ≡ (& a 1 - (! a a trunc ≡ ##a.T a - a 0.5 < a 1 - cases : ##fct 1.5 a 165 - < (a 165 > (| 1 cases : ##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 ##fct /^e ##g -^2 π * sqrt * ##x * (1 ##fct /^* (##fct^: π π a * sin ##S * / a 0.5 < ##S cases 71 line cases 11 line : 21 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 ##SS x a^* e x -^* : 31 line :

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

x Γ

4 appVersion 0.99.7822.147 # @ = Applications 7 s ≡ 21 mat

ε 1017 -^:

Sanity check Mathcad 11

a 0.5 :

x λ 10 x ≤ (x 2 a * ≤ (& cases :

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

1 ε Γ a ε Γ 21 mat 1.000000000000000 1.772453844580960 21 mat

Jnct a a Γ a a γ + 2 / : 0.8862

X 0.1 a 0.11 range : i 1 X rows range Y i el a X i el Γ : for x f X Y x cinterp : 31 line

0.5 0.5 γ 0.5 0.5 Γ 14.5 Γ 31 mat 1.21003561931112 0.5624182315944 2.30923179223141 1010^* 31 mat

U x f Jnct - 0 ≡ x X 1 el a solve : 0.227468207128136

NOTE ... Smath plots at the kernel level
but solves neither γ(a,x), Γ(a,x).
It complains about 'k'...
f(x) reconciliates the solver.

a x Γ x λ * a x γ x λ * U a U Γ + 50 black 1 5 mat a a a Γ . 14 blue 1 5 mat a a a γ . 14 red 1 5 mat 5 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

Example_2: Fitting Mathsoft 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 :

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 : n 2.5 : 14 mat

x f yo a b x Γ n^* + :

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 :

pts Ω 2 1 sys

yo 10.028 : a 143.017 : b 1.7666 : n 2.486 : 14 mat

1 time t0 - 7.8 s *

a 0.5 :

"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 :

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 :

sol 11015 -^* 0.238739753660939 0.23385967247468 0.227353946214894 0.227469375974804 0.227468211772556 0.227468211559788 0.227468211559788 81 mat

0.22746820712813