I need to solve an equation containing the Sine Integral $\mathrm{Si}\left(\frac{2 k \pi}{x}\right)$ in mathjs which doesn't have the $\mathrm{Si}$ function. Is there another way to represent this?
If $$
\mathrm{Si}\left(z\right) = \int_{0}^{z}{\frac{\sin{t}}{t}\,\mathrm{d}t}
$$
How do I actually calculate $\mathrm{Si}\left(…\right)$. It seems like I have to find a way to integrate $z$ every time I see $\mathrm{Si}\left(z\right)$ but calculators and computers wouldn't do that if $\mathrm{Si}\left(z\right)$ is a known function?
See : https://www.wolframalpha.com/input/?i=integrate+sin%5E2%281+%2F+x%29
Best Answer
If you want to compute $$\begin{align} \operatorname{Si}(x) &= \int_0^x \frac{\sin t}t dt , \end{align}$$ for $0\leq x \leq \pi$, you could use the magnificent approximation $$\sin(t) \sim \frac{16 (\pi -t) t}{5 \pi ^2-4 (\pi -t) t}\qquad (0\leq t\leq\pi)$$ proposed, more than $\color{red}{1400}$ years ado by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician.
If you think about it, it is a kind of Padé approximant.
As a result, this will give the simple $$\operatorname{Si}(x)\sim -2 \left(\log \left(\frac{4 x^2}{5 \pi ^2}-\frac{4 x}{5 \pi }+1\right)+\tan ^{-1}\left(\frac{4 x}{2 x-5 \pi }\right)\right) $$ which shows a maximum absolute error of $0.00367$ and a maximum relative error of $1.86$%.
Much better would be the $[7,6]$ Padé approximant which I shall write as $$\operatorname{Si}(x)\sim x \,\frac{1+\sum _{i=1}^3 a_i\,x^{2 i} } {1+\sum _{i=1}^3b_i\,x^{2 i} }$$ where the $a_i$'s and $b_i$'s are respectively $$\left\{-\frac{13524601565}{379956015036},\frac{567252710471}{766244630322600},- \frac{35803984658017}{8109933167334398400}\right\}$$ $$\left\{\frac{842673993}{42217335004},\frac{1864994705}{10216595070968},\frac{532 2538193}{6620353605987264}\right\}$$ which gives a maximum absolute error of $5.21 \times 10^{-7}$.