Generalized Fresnel integrals

Generalized Fresnel integrals ( Boehmer integrals ) are special functions generalizing Fresnel integrals . Introduced by Peter Böhmer in 1939 ^[1] .

The generalized Fresnel cosine:

\operatorname {C} (x,y)=\int _{x}^{\infty }t^{y-1}\cos(t)\,dt

{\ displaystyle \ operatorname {C} (x, y) = \ int _ {x} ^ {\ infty} t ^ {y-1} \ cos (t) \, dt}

{\ displaystyle \ operatorname {C} (x, y) = \ int _ {x} ^ {\ infty} t ^ {y-1} \ cos (t) \, dt}

Generalized Fresnel sine:

\operatorname {S} (x,y)=\int _{x}^{\infty }t^{y-1}\sin(t)\,dt

{\ displaystyle \ operatorname {S} (x, y) = \ int _ {x} ^ {\ infty} t ^ {y-1} \ sin (t) \, dt}

{\ displaystyle \ operatorname {S} (x, y) = \ int _ {x} ^ {\ infty} t ^ {y-1} \ sin (t) \, dt}

Accordingly, the usual Fresnel integrals are expressed in terms of the Böhmer integrals as follows:

\operatorname {S} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {S} \left({\frac {1}{2}},y^{2}\right)

{\ displaystyle \ operatorname {S} (y) = {\ frac {1} {2}} - {\ frac {1} {\ sqrt {2 \ pi}}} \ cdot \ operatorname {S} \ left ({ \ frac {1} {2}}, y ^ {2} \ right)}

{\ displaystyle \ operatorname {S} (y) = {\ frac {1} {2}} - {\ frac {1} {\ sqrt {2 \ pi}}} \ cdot \ operatorname {S} \ left ({ \ frac {1} {2}}, y ^ {2} \ right)}

\operatorname {C} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {C} \left({\frac {1}{2}},y^{2}\right)

{\ displaystyle \ operatorname {C} (y) = {\ frac {1} {2}} - {\ frac {1} {\ sqrt {2 \ pi}}} \ cdot \ operatorname {C} \ left ({ \ frac {1} {2}}, y ^ {2} \ right)}

{\ displaystyle \ operatorname {C} (y) = {\ frac {1} {2}} - {\ frac {1} {\ sqrt {2 \ pi}}} \ cdot \ operatorname {C} \ left ({ \ frac {1} {2}}, y ^ {2} \ right)}

Also, through the generalized Fresnel integrals, we can express the integral sine and integral cosine :

\operatorname {Si} (x)={\frac {\pi }{2}}-\operatorname {S} (x,0)

{\ displaystyle \ operatorname {Si} (x) = {\ frac {\ pi} {2}} - \ operatorname {S} (x, 0)}

{\ displaystyle \ operatorname {Si} (x) = {\ frac {\ pi} {2}} - \ operatorname {S} (x, 0)}

\operatorname {Ci} (x)={\frac {\pi }{2}}-\operatorname {C} (x,0)

{\ displaystyle \ operatorname {Ci} (x) = {\ frac {\ pi} {2}} - \ operatorname {C} (x, 0)}

{\ displaystyle \ operatorname {Ci} (x) = {\ frac {\ pi} {2}} - \ operatorname {C} (x, 0)}

Literature

KB Oldham, JC Myland, J. Spanier. An atlas of functions . - 2nd ed. - Springer, 2008 .-- 748 p.

Notes

↑ PE Böhmer. Differenzengleichungen und bestimmte Integrale (German) . - Leipzig, KF Koehler Verlag, 1939 .-- 148 p.

[1] PE Böhmer. Differenzengleichungen und bestimmte Integrale (German) . - Leipzig, KF Koehler Verlag, 1939 .-- 148 p.

Generalized Fresnel integrals

Literature

Notes

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