Böhmer integral
In mathematics, a Böhmer integral is an integral introduced by (Böhmer 1939) generalizing the Fresnel integrals. There are two versions, given by [math]\displaystyle{ \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} }[/math]
Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as [math]\displaystyle{ \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} }[/math]
The sine integral and cosine integral can also be expressed in terms of the Böhmer integrals [math]\displaystyle{ \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} }[/math]
References
- Böhmer, Paul Eugen (1939) (in German). Differenzengleichungen und bestimmte Integrale.. Leipzig, K. F. Koehler Verlag. https://books.google.com/books?id=pD5tAAAAMAAJ.
- Oldham, Keith B.; Myland, Jan; Spanier, Jerome (2010). An Atlas of Functions. Springer Science & Business Media. p. 401. ISBN 9780387488073. https://books.google.com/books?id=UrSnNeJW10YC&pg=PA401.
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