Böhmer integral

In mathematics, a Böhmer integral is an integral introduced by Template:Harvtxt generalizing the Fresnel integrals.

There are two versions, given by $$\begin{array}{cc}\mathrm{C}(x,\alpha )& =\underset{x}{\overset{\mathrm{\infty }}{\int }}{t}^{\alpha -1}\cos (t)dt\\ \mathrm{S}(x,\alpha )& =\underset{x}{\overset{\mathrm{\infty }}{\int }}{t}^{\alpha -1}\sin (t)dt\end{array}$$

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

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

References

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