Böhmer integral

In mathematics, a Böhmer integral is an integral introduced by Böhmer (1939) generalizing the Fresnel integrals.

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

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

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