Definition:Fresnel Integral/Sine

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Definition

The Fresnel sine integral function is the real function $\operatorname S: \R \to \R$ defined by:

$\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$


Asymptotic Expansion for Fresnel Sine Integral Function

$\map {\mathrm S} x \sim \dfrac 1 2 - \dfrac 1 {\sqrt {2 \pi} } \paren {\map \cos {x^2} \paren {\dfrac 1 x - \dfrac {1 \times 3} {2^2 x^5} + \dfrac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} + \map \sin {x^2} \paren {\dfrac 1 {2 x^3} - \dfrac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }$


Also defined as

The Fresnel sine integral function can also be seen defined as:

$\ds \map {\operatorname S} x = \int_0^x \sin u^2 \rd u$


In the context of physics, the Fresnel sine integral function is most often defined as:

$\ds \map {\operatorname S} x = \int_0^x \map \sin {\dfrac {\pi u^2} 2} \rd u$


Also see

  • Results about the Fresnel sine integral function can be found here.


Source of Name

This entry was named for Augustin-Jean Fresnel.


Historical Note

The Fresnel integrals were used by Augustin-Jean Fresnel to analyse the diffraction of light.


Linguistic Note

The eponym Fresnel is pronounced fre-nell.


Sources