NOTES about integration

8955f256-d61d-4809-9733-11aac14f4bf0 46 4 5 decimal

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

Si(x), Ci(x) ... Integral representation ... acc 500

datum π 2 / :

Euler constant γ 0.57721566490153286 :

▼ Transit ▼

Si(x)... free of freak z Ci ε 1015 -^: z Ci γ x ln + t cos 1 - t / t ε x int + : 21 line :

Si(x)... free of freak z Si ε 1015 -^: z Si t sin t / t ε x int : 21 line :

x Ci z Ci :

x Si z Si :

1 Ci 0.3374

1 Si 0.9461

x plot x Si x Ci datum 41 sys :

Complete Integral representation

x plot

sanity Abramowitz  p. 242
...  acc 500 7 Si 7 Ci 21 mat 1.4545966142 0.0766952785 21 mat

NOTES about integration

ε 1015 -^:

16.3658005743936
very accurate
... acc 1000 x Si x ε 11 int 16.3658005743936

Does NOT integrate
... acc 2000 [33 min]
0.014795716100422 MCD x Ci x ε 11 int 0.00914 -

Integrate over Σ(n)
∫kernel(f,ε,11,20000) 0.014795722454334

Si(x), Ci(x) ... Σ(x) ... Σ(x,n) series representation

◄ CosineIntegral x Ci γ x ln + x 2^- (k^(2 k * 2 k * (! * / k 140 sum + :

SineIntegral ► x Si 1 - (m^x 12 m * +^* 12 m * + (12 m * + (! (* / (m 040 sum :

x Ci 0 ≡ x 0.5 0.75 solve 0.6165

optional ON/OFF shift 0.6165 :

Nielsen_0 x 0.1 20 0.2 range : i 1 x rows range Ux i el x i el Si eval : Vy i el x i el Ci eval : 21 line for Ux Vy augment 31 line :

Nielsen_1 x 0.6165 20 0.7 range : i 1 x rows range Nx i el x i el Si eval : Ny i el x i el Ci eval : 21 line for Nx shift - Ny augment 31 line :

usable length x 20 Si 16 ≤ x 40 Si 32 ≤ x 80 Si 40 ≤ 31 mat

Nielsen_0 Nielsen_0 :

Nielsen_1 Nielsen_1 :

lookup(value,table,col)

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

ROW Nielsen_0 2 col Max Nielsen_0 2 lookup : 1611 mat

Nielsen_0 ROW row 1.3892 0.4717 12 mat

Nielsen_0 1.3892 0.4717 + 30 red 1 5 mat 2 1 sys Nielsen_1 2 1 sys

...  Abramowitz  p. 242 7 Si 7 Ci 21 mat 1.4545966142 0.0766952785 21 mat

∫kernel(f,a,b,n)

f a b n ∫kernel 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 / : 14 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 * + : 32 / b a - (* Simpson * 41 line :

x f x Ci :

0.014795716100422 ... MCD 11 f ε 1120000 ∫kernel 0.014795722454334

x f x Si :

16.3658005743911 ... MCD 11 f ε 111000 ∫kernel 16.3658005743862

x f x Si :

TOL 106 -^:

Golden-Ratio search [min/max]

Golden ratio

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

X f 24 y1 y2 ≥ TOL Extrema 1 el f 57 y1 y2 ≤ TOL Extrema 1 el 21 mat :

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

X 1 el f X 2 el f 21 mat 1.852 1.418 21 mat

π
2π X 3.1416 6.2832 21 mat

datum 8 - datum 32 datum 22 mat :

x λ 132 - x ≤ (x 32 ≤ (& cases :

x Si x λ * datum X 1 el X 1 el f + 20 black 1 5 mat X 2 el X 2 el f + 20 black 1 5 mat 4 1 sys

max deviation X 1 el f X 2 el f - abs 0.4338

Nielsen_0 rows 199

τ 1190 range :

t NX Nielsen_0 1 col t el : t NY Nielsen_0 2 col t el : 12 mat

t 1 ≡ path t NX t NY 1 2 mat : path path t NX t NY 1 2 mat stack : if 1.3892 0.4717 + 30 red 1 5 mat 2 1 sys

SineIntegral ► x Si 1 - (n^x 12 n * +^* 12 n * + (12 n * + (! (* / (n 040 sum :

x λ 10 x ≤ (x 32 ≤ (& cases :

datum π 2 / :