In mathematics, a Böhmer integral is an integral introduced by Böhmer (1939) generalizing the Fresnel integrals. There are two versions, given by
C ( x , α ) = ∫ x ∞ t α − 1 cos ( t ) d t S ( x , α ) = ∫ x ∞ t α − 1 sin ( t ) d t {\displaystyle {\begin{aligned}\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{aligned}}}
Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as
S ( y ) = 1 2 − 1 2 π ⋅ S ( 1 2 , y 2 ) C ( y ) = 1 2 − 1 2 π ⋅ C ( 1 2 , y 2 ) {\displaystyle {\begin{aligned}\operatorname {S} (y)&={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {S} \left({\frac {1}{2}},y^{2}\right)\\[1ex]\operatorname {C} (y)&={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {C} \left({\frac {1}{2}},y^{2}\right)\end{aligned}}}
The sine integral and cosine integral can also be expressed in terms of the Böhmer integrals
Si ( x ) = π 2 − S ( x , 0 ) Ci ( x ) = π 2 − C ( x , 0 ) {\displaystyle {\begin{aligned}\operatorname {Si} (x)&={\frac {\pi }{2}}-\operatorname {S} (x,0)\\[1ex]\operatorname {Ci} (x)&={\frac {\pi }{2}}-\operatorname {C} (x,0)\end{aligned}}}
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