Böhmer integral

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

There are two versions, given by

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

Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as

$${\begin{aligned}\operatorname {S} (y)&={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {S} \left({\frac {1}{2}},y^{2}\right)\\[1ex]\operatorname {C} (y)&={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {C} \left({\frac {1}{2}},y^{2}\right)\end{aligned}}$$

The sine integral and cosine integral can also be expressed in terms of the Böhmer integrals

$${\begin{aligned}\operatorname {Si} (x)&={\frac {\pi }{2}}-\operatorname {S} (x,0)\\[1ex]\operatorname {Ci} (x)&={\frac {\pi }{2}}-\operatorname {C} (x,0)\end{aligned}}$$

References

• Böhmer, Paul Eugen (1939). Differenzengleichungen und bestimmte Integrale (in German). Leipzig, K. F. Koehler Verlag.
• Oldham, Keith B.; Myland, Jan; Spanier, Jerome (2010). An Atlas of Functions. Springer Science & Business Media. p. 401. ISBN 9780387488073.