Special function defined by an integral
Plots of S (x )C (x )C (x )451 424 S and C were defined using π / 2 t 2 t 2 The Fresnel integrals S (x )C (x )transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf ). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral 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 parametric curve
(
S
(
t
)
,
C
(
t
)
)
{\displaystyle {\bigl (}S(t),C(t){\bigr )}}
is the Euler spiral 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 Cauchy's integral theorem .
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