Trigonometric integrals

Summary

The trigonometric integrals are the sine integral $\operatorname{Si}(x) = \int_0^x \frac{\sin t}{t}\, dt$ and the cosine integral $\operatorname{Ci}(x) = -\int_x^\infty \frac{\cos t}{t}\, dt$, with their hyperbolic versions $\operatorname{Shi}$ and $\operatorname{Chi}$. They are antiderivatives of $\frac{\sin x}{x}$ and $\frac{\cos x}{x}$, which have no elementary antiderivatives. The sine integral tends to $\frac{\pi}{2}$, the value of the Dirichlet integral, and its overshoot describes the Gibbs phenomenon of Fourier series. They are closely related to the exponential integral.