5ab00e81-beea-4f36-a37f-3e37937650e8 7 6 9 decimal

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

Incremental analysis procedure

SineIntegral x Si 1 - (n^x 12 n * +^* 12 n * + (12 n * + (! (* / (n 070 sum eval :

This particular Si(x) fails both operators [Derivative/Integrate], maybe same to generalised power series ? For applying "Incremental Analysis", isolate the components respectively.

x f 11 x 2^+ / : User function ≡

Algo style Memo formula of incremental analysis of this demo example h 0.00001 : increment ≡ x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - : 1rst Derivative ≡ x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : 2nd Derivative ≡ x h f h_int h x h + f x f - (* 2 x f * + (x * : Integral Curve [antiderivative] ≡ x h f h_cumul 2 h x h + f x f - (* 2 x f * + (x 2^* (* : Cumulative Integration ≡ 51 line

x f x h f D1 x h f D2 x h f h_int 0.5 x h f h_cumul * 5 1 sys x Si 8 8 Si + 15 red 1 5 mat 2 1 sys

0.1 Si #

Sanity [N = 40] 0.1 Si 0.099944461 ≡ sanity ≡

x Si 1 1 sys

MedCalc

https://www.medcalc.org/manual/sinintegral_function.php

Cumulative integation- Trapeze Method f a b N NIntTrap h b a - N / : h 2 / a f b f + 2 a k h * + f k 1 N 1 - sum * + (* 21 line :

Si 1 - 1100 NIntTrap 1.570763

Isolate components

x f x Si :

h 0.0001 :

x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - eval :

x h f h_int h x h + f x f - (* 2 x f * + (x * eval :

Si(x), Deriv[Si(x)], Integral curve[Si(x)]

x Si x h f D1 0.05 x h f h_int * 3 1 sys