Böhmer integral

In mathematics, a Böhmer integral is an integral introduced by (Böhmer 1939) generalizing the Fresnel integrals. There are two versions, given by $$\begin{array}{rl}\mathrm{C}(x,\alpha )& =\underset{x}{\overset{\mathrm{\infty }}{\int }}{t}^{\alpha -1}\cos (t)dt\\ \mathrm{S}(x,\alpha )& =\underset{x}{\overset{\mathrm{\infty }}{\int }}{t}^{\alpha -1}\sin (t)dt\end{array}$$

Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as $$\begin{array}{rl}\mathrm{S}(y)& =\frac{1}{2}-\frac{1}{\sqrt{2\pi }}\cdot \mathrm{S}(\frac{1}{2},y^2)\\ \mathrm{C}(y)& =\frac{1}{2}-\frac{1}{\sqrt{2\pi }}\cdot \mathrm{C}(\frac{1}{2},y^2)\end{array}$$

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

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.