There are a number of slightly different definitions of the Fresnel integrals. In physics, the Fresnel integrals denoted 
and 
are most often defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
so
 |  |  |
(3)
|
 |  |  |
(4)
|
These Fresnel integrals are implemented in the Wolfram Language as FresnelC[z] and FresnelS[z].
and 
are entire functions.
The 
and 
integrals are illustrated above in the complex plane.
They have the special values
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
and
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
An asymptotic expansion for 
gives
 |  |  |
(11)
|
 |  |  |
(12)
|
Therefore, as 
,
and 
.
The Fresnel integrals are sometimes alternatively defined as
 |  |  |
(13)
|
 |  |  |
(14)
|
Letting 
so 
,
and 
 |  |  |
(15)
|
 |  |  |
(16)
|
In this form, they have a particularly simple expansion in terms of spherical Bessel functions of the first kind. Using
 |  |  |
(17)
|
 |  |  |
(18)
|
 |  |  |
(19)
|
where 
is a spherical Bessel function
of the second kind
 |  |  |
(20)
|
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
 |  |  |
(24)
|
Related functions 
,
,
,
and 
are defined by
 |  |  |
(25)
|
 |  |  |
(26)
|
 |  |  |
(27)
|
 |  |  |
(28)
|
and 
."
§1.3 in 
and 
."
Ch. 39 in