Böhmer Integral

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

There are two versions, given by
\begin
\operatorname(x,\alpha) &= \int_x^\infty t^ \cos(t) \, dt \\ ex\operatorname(x,\alpha) &= \int_x^\infty t^ \sin(t) \, dt
\end

Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as
\begin
\operatorname(y) &= \frac1-\frac1\cdot\operatorname\left(\frac1,y^2\right) \\ ex\operatorname(y) &= \frac1-\frac1\cdot\operatorname\left(\frac1,y^2\right)
\end

The sine integral and cosine integral can also be expressed in terms of the Böhmer integrals
\begin
\operatorname(x) &= \frac - \operatorname(x,0) \\ ex\operatorname(x) &= \frac -\operatorname(x,0)
\end

References

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Special functions

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