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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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c673c7b46d9fb39098030b97522d20ad3507cc88)
![{\displaystyle \displaystyle S(x,\alpha )=\int _{x}^{\infty }t^{\alpha -1}\sin(t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fad18dab4153897a1f5d72adae0a70d8a063109c)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6d1481286cb63d9e382338c9ad2364d973d704)
![{\displaystyle \operatorname {C} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {C} \left({\frac {1}{2}},y^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/410c1f89d8db845896237631364318bc32a1e44b)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d3d8ce93c681118dcc8e4116ee57697d40810e)
![{\displaystyle \operatorname {Ci} (x)={\frac {\pi }{2}}-\operatorname {C} (x,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80d76c0cf01ee189aaec11e1c2b21b8216e5ffaf)
References