<p> Γ(x) [15 D] Alvaro Same otherwise </p>

ea058747-1f9b-406a-a89a-5add6651add5 35 4 5 decimal

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

al_airy("number") - (x) Returns the Airy functions: Ai(x), Ai'(x), Bi(x) and Bi'(x). Ai(x) Ai'(x) Bi(x) Bi'(x)

Ai
.....
Ai'
.....
Bi
.....
Bi' Airy 1 Airy rows 5 range 12 range el Airy 2 Airy rows 6 range 12 range el Airy 3 Airy rows 7 range 12 range el Airy 4 Airy rows 8 range 12 range el 41 sys

0 al_airy 0.355 0.2588 - 0.6149 0.4483 41 mat

t0 1 time :

Airy data vector of nested quadruplets al_airy(v) 21 sys v 10 - π 9.95 - range : data v al_airy vectorize eval : unest the quadruplets v data 1 el : j 2 data rows range v v data j el stack : for 21 line v j 1 v rows range : j v augment 81 line :

Airy Airy eval :

n Airy rows :

sub Airy 1 n 5 range 12 range el :

1 time t0 - 19 s *

sub

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 0 : H π : N sub rows : 13 mat X L H N xd : 21 line

faster SYMBOLIC scalar X sub 2 col augment :

x Airy scalar 1 col scalar 2 col x cinterp :

x λ 10 x ≤ (x π ≤ (& cases :

Scalar wrt 'x' from cinterp(,,,)

x Airy x λ * π 0.5 π 0.5 - 2 2 mat 2 1 sys

Not scalar wrt 'x' but on range

scalar π 0.5 π 0.5 - 2 2 mat 2 1 sys

MCD 11 ... 0.331 x Airy x 0 π int 0.263

A Airy 1 n 5 range 12 range el Airy 2 n 6 range 12 range el 21 sys :

B Airy 3 n 7 range 12 range el Airy 4 n 8 range 12 range el 21 sys :

A 1 el A 2 el n 1 n 1 - 2 2 mat 3 1 sys B 1 el B 2 el n 10 n 4 - 2 2 mat 3 1 sys

1 time t0 - 25 s *

sanity Mathcad 11 10 - al_airy 0.040241 0.996265 0.31468 - 0.119414 41 mat

4 appVersion 0.98.6179.21440 ≡

The incomplete beta function arises often in probabilisitic applications

<p> Γ(x) [15 D] Alvaro Same otherwise </p>

Γ 0.99999999999999709182 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 -^* 151 mat :

ζ Γ z ζ 1 - : zh z 0.5 + : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ 1 el Γ κ 1 + el z κ + / κ 114 sum + (* zp 2^* zg - exp * eval 21 line :

Valid for Re(ζ)>0

x y + Γ x Γ y Γ * / t x 1 -^1 t - (y 1 -^* t 0 a int *

Mathcad from integration 0.2345 12 ibeta 0.414009750000000 ≡

Mathcad from built-in 0.2345 12 ibeta 0.414009750000000 ≡

Smath 4 decimals x Γ x Gamma :

a x y f x y + Γ x Γ y Γ * / t x 1 -^1 t - (y 1 -^* t 0 a int * :

option 100 0.2345 12 f 0.414009750000000

Airy data vector of nested quadruplets al_airy(v) 21 sys v 10 - π 9.95 - range : data v al_airy vectorize eval : unest the quadruplets v data 1 el : j 2 data rows range v v data j el stack : for 21 line v j 1 v rows range : j v augment 81 line :

Airy Airy eval :

sub1 Airy 1 n 5 range 12 range el :

sub2 Airy 2 n 6 range 12 range el :

sub3 Airy 3 n 7 range 12 range el :

sub4 Airy 4 n 8 range 12 range el :

sub1 sub2 sub3 sub4 4 1 sys

inaccurate from Smath builtin Gamma(x) 0.2345 12 BetaRegularized 0.41400908578928

x 10 - π 9.95 - range :

A x x al_airy vectorize augment :

Ai
.....
Ai'
.....
Bi
.....
Bi' sub Airy 1 n 5 range 12 range el Airy 2 n 6 range 12 range el Airy 3 n 7 range 12 range el Airy 4 n 8 range 12 range el 41 sys :

A 10 - 0.0402 0.9963 0.3147 - 0.1194 41 mat 9.95 - 0.0893 0.9639 0.3048 - 0.2742 41 mat 9.9 - 0.1362 0.9078 0.2874 - 0.4215 41 mat 9.85 - 0.1798 0.8297 0.2629 - 0.5576 41 mat 9.8 - 0.2189 0.7315 0.2319 - 0.6794 41 mat 52 mat

sanity Mathcad 11 10 - al_airy 0.040241 0.996265 0.31468 - 0.119414 41 mat

sub sys2mat rows 4

Airy sub sys2mat 1 el 11 mat sub sys2mat 2 el 11 mat sub sys2mat 3 el 11 mat sub sys2mat 4 el 11 mat augment :

Spool to file ... Export

CWD C:\SmathFile CurrentDirectory :

Airy CWD Airy wfile 1

airy CWD Airy rfile :