Special function defined by an integral
Plots of S (x ) and C (x ) . The maximum of C (x ) is about 0.977451 424 . If the integrands of S and C were defined using π / 2 t 2 instead of t 2 , then the image would be scaled vertically and horizontally (see below).
The Fresnel integrals S (x ) and C (x ) are two 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.