Böhmer integral

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

There are two versions, given by

${\displaystyle \displaystyle C(x,\alpha )=\int _{x}^{\infty }t^{\alpha -1}\cos(t)\,dt}$

${\displaystyle \displaystyle S(x,\alpha )=\int _{x}^{\infty }t^{\alpha -1}\sin(t)\,dt}$

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

${\displaystyle \operatorname {S} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {S} \left({\frac {1}{2}},y^{2}\right)}$

${\displaystyle \operatorname {C} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {C} \left({\frac {1}{2}},y^{2}\right)}$

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

${\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-\operatorname {S} (x,0)}$

${\displaystyle \operatorname {Ci} (x)={\frac {\pi }{2}}-\operatorname {C} (x,0)}$

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.