Fixed-lower-limit sine-integral expansion
= Fixed-lower-limit sine-integral expansion
For
$$
\operatorname{si}(x)=\int_1^\infty\frac{\sin(xt)}t\,dt,
$$
repeated integration by parts gives
$$
\operatorname{si}(x)\sim
\cos x\sum_{n\geq0}(-1)^n(2n)!x^{-2n-1}
+\sin x\sum_{n\geq0}(-1)^n(2n+1)!x^{-2n-2}.
$$