Böhmer integral

In mathematics, a Böhmer integral is an integral introduced by generalizing the Fresnel integrals.

There are two versions, given by $$\begin{align} \operatorname{C}(x,\alpha) &= \int_x^\infty t^{\alpha-1} \cos(t) \, dt \\[1ex] \operatorname{S}(x,\alpha) &= \int_x^\infty t^{\alpha-1} \sin(t) \, dt \end{align}$$

Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as $$\begin{align} \operatorname{S}(y) &= \frac1{2}-\frac1{\sqrt{2\pi}}\cdot\operatorname{S}\left(\frac1{2},y^2\right) \\[1ex] \operatorname{C}(y) &= \frac1{2}-\frac1{\sqrt{2\pi}}\cdot\operatorname{C}\left(\frac1{2},y^2\right) \end{align}$$

The sine integral and cosine integral can also be expressed in terms of the Böhmer integrals $$\begin{align} \operatorname{Si}(x) &= \frac{\pi}{2} - \operatorname{S}(x,0) \\[1ex] \operatorname{Ci}(x) &= \frac{\pi}{2} -\operatorname{C}(x,0) \end{align}$$