Fresnel integral

The Fresnel integrals S(x) and C(x) are two <a href="/facts/Transcendental_function/yMkeq4Eo">transcendental functions</a> named after <a href="/facts/Augustin-Jean_Fresnel/jb64tyGc">Augustin-Jean Fresnel</a> that are used in <a href="/facts/Optics/p0vq4S1T">optics</a> and are closely related to the <a href="/facts/Error_function/Ca2H7pGr">error function</a> (erf). They arise in the description of <a href="/facts/Near_and_far_field/wS2Dp5lv">near-field</a> <a href="/facts/Fresnel_diffraction/lxyq42M4">Fresnel diffraction</a> phenomena and are defined through the following <a href="/facts/Integral/V7lyV997">integral</a> representations:

 
 
 
 S
 (
 x
 )
 =
 
 ∫
 
 0
 
 
 x
 
 
 sin
 ⁡
 
 (
 
 t
 
 2
 
 
 )
 
 
 d
 t
 ,
 
 C
 (
 x
 )
 =
 
 ∫
 
 0
 
 
 x
 
 
 cos
 ⁡
 
 (
 
 t
 
 2
 
 
 )
 
 
 d
 t
 .
 
 
 {\displaystyle S(x)=\int _{0}^{x}\sin \left(t^{2}\right)\,dt,\quad C(x)=\int _{0}^{x}\cos \left(t^{2}\right)\,dt.}

The <a href="/facts/Parametric_curve/5Uvrlgqf">parametric curve</a> ⁠
 
 
 
 
 
 (
 
 
 S
 (
 t
 )
 ,
 C
 (
 t
 )
 
 
 )
 
 
 
 
 {\displaystyle {\bigl (}S(t),C(t){\bigr )}}
 
⁠ is the <a href="/facts/Euler_spiral/umcvnb2J">Euler spiral</a> or clothoid, a curve whose curvature varies linearly with arclength.
The term Fresnel integral may also refer to the complex definite integral

 
 
 
 
 ∫
 
 −
 ∞
 
 
 ∞
 
 
 
 e
 
 ±
 i
 a
 
 x
 
 2
 
 
 
 
 d
 x
 =
 
 
 
 π
 a
 
 
 
 
 e
 
 ±
 i
 π
 
 /
 
 4
 
 
 
 
 {\displaystyle \int _{-\infty }^{\infty }e^{\pm iax^{2}}dx={\sqrt {\frac {\pi }{a}}}e^{\pm i\pi /4}}

where a is real and positive; this can be evaluated by closing a contour in the complex plane and applying <a href="/facts/Cauchy%2527s_integral_theorem/NIgLv1VB">Cauchy's integral theorem</a>.

Fresnel integral open-in-new

Fresnel integral