Γ(x) [15 D] Alvaro Same otherwise

359ffc79-a055-49da-b67a-7dcd1a7a0cd5 11 4 5 decimal

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

Whittaker reconciliation wrt Simpson integrator

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

Whittaker 2nd kind
acc 100 ~ convenient
this demo ▼ 5 min k m x W x k^x 2 / - exp * m k - 12 / + Γ / t - exp t m k - 12 / - (^* 1 t x / + (m k + 12 / - (^* t 0.005 20 int * :

x λ 10 x ≤ (x 10 ≤ (& cases :

@ accuracy 10000 ...won't end !

0.5 0.125 x W x λ * 0.5 1 x W x λ * 1 0.6 x W x λ * 0.5 0.5 x W x λ * 4 1 sys

0.5 25 W 0.5 15 W 0.5 0.5 5 W 0.25 25 W 1 0.75 5 W 0.5 0.125 1 W 0.5 0.95 1 W 71 mat 0.374 0.219 0.198 0.242 0.427 0.45 1.135 71 mat

Whittaker reconciliation wrt Adaptive Integrator ▼

∫Adaptive(f,a,b,N,ε) Adapted 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 :

Γ(x) [15 D] Alvaro Same otherwise

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 λ 10 x ≤ (x 10 ≤ (& cases :

k 0.5 : m 0.125 : 12 mat k m x f x - exp x m k - 12 / - (^* 1 x x / + (m k + 12 / - (^* eval : x f k m x f : N 100 : low 1012 -^: ε ε 1015 -^: (: 13 mat x W x k^x 2 / - exp * m k - 12 / + Γ / f low x N ε ∫Adaptive * : 51 sys

Steps to Adaptive ... 1. plot the integrand f(k,m,x) 2. By inspection, determine if ? Simpson/Adaptive ? 3. Adaptive demo takes about 7 min. 4. Mathcad Adaptive is not absolute Bible, reputed reliable

t0 1 time :

0.4189

1 W 0.6188

0.5161

1 time t0 - min 0

0.5474

0.5884

k m x f

k m low f 3.4485 1010^*

maple 0.641 ≡

0.6188

0.5 25 W 0.5 15 W 0.5 0.5 5 W 0.25 25 W 1 0.75 5 W 0.5 0.125 1 W 0.5 0.95 1 W 71 mat

x λ 10 x ≤ (x 10 ≤ (& cases :

k 0.5 : m 0.125 : 12 mat k m x f x - exp x m k - 12 / - (^* 1 x x / + (m k + 12 / - (^* eval : x f k m x f : 31 sys

copy/paste to isolate # # int

x f x int maple 828 nthroot * x 2 / - exp * 8 x * 116 / 916 / x WhittakerM * 91716 / 916 / x WhittakerM * + (* 9 x 16 nthroot 15^* /

∫Adaptive(f,a,b,N,ε) Adapted 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 :

Γ(x) [15 D] Alvaro Same otherwise

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 λ 10 x ≤ (x 10 ≤ (& cases :

k 0.5 : m 0.125 : 12 mat k m x f x - exp x m k - 12 / - (^* 1 x x / + (m k + 12 / - (^* eval : x f k m x f : N 100 : low 1012 -^: ε ε 1015 -^: (: 13 mat x W x k^x 2 / - exp * m k - 12 / + Γ / f low x N ε ∫Adaptive * : 51 sys

0.4189

0.5161

1 W 0.6188

0.5474

0.5884

0.5 0.125 x f

0.6188

k m low f 3.4485 1010^*

Γ(x) [15 D] Alvaro Same otherwise

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 λ 10 x ≤ (x 10 ≤ (& cases :

k 0.5 : m 0.125 : 12 mat k m x f x - exp x m k - 12 / - (^* 1 x x / + (m k + 12 / - (^* eval : x f k m x f : x W x k^x 2 / - exp * m k - 12 / + Γ / x f * : 41 sys

0.5 25 W 0.5 15 W 0.5 0.5 5 W 0.25 25 W 1 0.75 5 W 0.5 0.125 1 W 0.5 0.95 1 W 71 mat

0.001 f 459.4037

x k ^ x 2 / - exp * m k - 1 2 / + Γ / x f * x f x λ *

copy/paste to isolate # # int

x k^x 2 / - exp * m k - 12 / + Γ / x f x int (* maple 8000000000000028 nthroot * x 2 / - exp 2^* 8 x * 116 / 916 / x WhittakerM * 91716 / 916 / x WhittakerM * + (* 678054743891781 x 16 nthroot 7^* /

x plot x k^x 2 / - exp * m k - 12 / + Γ / x f x int (* maple :

search manually the loer limit ▼ 0.0205 plot evalf maple 0.1007

6.5 plot evalf maple 0.1078

1 plot evalf maple 0.641

x plot

a:=plot(`in`, x=0..16, y=0..0.75, axes=boxed, color=blue, labels=["x","y "], tickmarks=[3,3]):c:=coordplot(rectangular, [-5..5, -1..1], color=[gray,gray], linestyle=[7,7], grid=[5,5]):display([a,c]);

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

L 0.021 : H 6.5 : N 45 : 13 mat

U L H N xd :

W j 1 U rows range Wh j el U j el plot evalf maple : for U Wh augment 21 line :

W 11 mat 0.021 0.1022 0.165 0.3397 0.309 0.4611 0.4529 0.5376 0.5969 0.5868 0.7409 0.6174 0.8849 0.6346 1.0288 0.6418 1.1728 0.6415 1.3168 0.6355 1.4608 0.6251 1.6048 0.6114 1.7487 0.5953 1.8927 0.5774 2.0367 0.5582 2.1807 0.5381 2.3246 0.5175 2.4686 0.4967 2.6126 0.4758 2.7566 0.455 2.9006 0.4345 3.0445 0.4144 3.1885 0.3948 3.3325 0.3756 3.4765 0.3571 3.6204 0.3392 3.7644 0.3219 3.9084 0.3052 4.0524 0.2892 4.1964 0.2739 4.3403 0.2593 4.4843 0.2452 4.6283 0.2318 4.7723 0.2191 4.9162 0.2069 5.0602 0.1954 5.2042 0.1844 5.3482 0.1739 5.4922 0.164 5.6361 0.1546 5.7801 0.1457 5.9241 0.1372 6.0681 0.1293 6.212 0.1217 6.356 0.1146 452 mat 11 mat