View source for Fresnel integrals

<!--
f0417201.png
$#A+1 = 26 n = 0
$#C+1 = 26 : ~/encyclopedia/old_files/data/F041/F.0401720 Fresnel integrals
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The special functions

$$ 
C ( x)  =  {
\frac{1}{\sqrt {2 \pi } }
 }
\int\limits _ { 0 } ^ { {x  ^ {2}} }

\frac{\cos  t }{\sqrt t }
  dt  = \ 
\sqrt {
\frac{2} \pi 
 }
\int\limits _ { 0 } ^ { x } 
\cos  t  ^ {2}  dt,
$$

$$ 
S ( x)  =  {
\frac{1}{\sqrt {2 \pi } }
 } \int\limits _ { 0 } ^ { {x } 
 ^ {2} } 
\frac{\sin  t }{\sqrt t }
  dt  =  \sqrt {
\frac{2} \pi 
 } \int\limits _ { 0 } ^ { x }  \sin  t  ^ {2}  dt;
$$

$$ 
\lim\limits _ {x \rightarrow + \infty }  C ( x)  =  \lim\limits _ {x \rightarrow + \infty }  S ( x)  =  {
\frac{1}{2}
 } .
$$

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041720a.gif" />

Figure: f041720a

The Fresnel integrals can be represented in the form of the series

$$ 
C ( x)  =  \sqrt {
\frac{2} \pi 
 } x
\sum _ {k = 0 } ^  \infty  

\frac{(- 1)  ^ {k} x  ^ {4k} }{( 2k)! ( 4k + 1) }
 ,
$$

$$ 
S ( x)  =  \sqrt {
\frac{2} \pi 
 } x \sum _ {k = 0 } ^  \infty   
\frac{(- 1)  ^ {k} x ^ {2 ( 2 k + 1) } }{( 2k + 1)! ( 4k + 3) }
 .
$$

An asymptotic representation for large  $  x $
is:

$$ 
C ( x)  =  {
\frac{1}{2}
 } +

\frac{1}{\sqrt {2 \pi } x }
 \ 
\sin  x  ^ {2} + O
\left ( 
\frac{1}{x  ^ {2} }
 \right ) ,
$$

$$ 
S ( x)  =  {
\frac{1}{2}
 } - 
\frac{1}{\sqrt {2 \pi } x }
  \cos
 x  ^ {2} + O \left ( 
\frac{1}{x  ^ {2} }
 \right ) .
$$

In a rectangular coordinate system  $  ( x, y) $
the projections of the curve

$$ 
x  =  t,\ \ 
y  =  C \left ( {
\frac \pi {2}
 } t  ^ {2} \right ) ,\ \ 
z  =  S \left ( {
\frac \pi {2}
 } t  ^ {2} \right ) ,
$$

where  $  t $
is a real parameter, onto the coordinate planes are the [[Cornu spiral|Cornu spiral]] and the curves  $  y = C ( \pi t  ^ {2} /2) $, 
$  z = S ( \pi t  ^ {2} /2) $(
see Fig. b).  $  x $
$  S( x) $
$  C( x) $

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041720b.gif" />

Figure: f041720b

The generalized Fresnel integrals (see [[#References|[1]]]) are functions of the form

$$ 
C ( x, \alpha )  = \ 
\int\limits _ { x } ^  \infty  
t ^ {\alpha - 1 } \ 
\cos  t  dt,
$$

$$ 
S ( x, \alpha )  =  \int\limits _ { x } ^  \infty   t ^ {\alpha - 1 }  \sin  t  dt.
$$

The Fresnel integrals are related to the generalized Fresnel integrals as follows:

$$ 
C ( x)  =  {
\frac{1}{2}
 } -

\frac{1}{\sqrt {2 \pi } }

C \left ( x  ^ {2} , {
\frac{1}{2}
 } \right ) ;
$$

$$ 
S ( x)  =  {
\frac{1}{2}
 } - 
\frac{1}{\sqrt {2 \pi } }
 S \left ( x  ^ {2} , {
\frac{1}{2}
 } \right ) .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Jahnke,   F. Emde,   "Tables of functions with formulae and curves" , Dover, reprint  (1945)  (Translated from German)</TD></TR></table>

====Comments====
A word of warning. There are different normalizations in use for the Fresnel integrals. E.g., in [[#References|[a3]]] they are defined as

$$ 
C _ {1} ( z)  =  \int\limits _ { 0 } ^ { z } 
\cos  
\frac{\pi t  ^ {2} }{2}
  d t ,\  S _ {1} ( z)  = \ 
\int\limits _ { 0 } ^ { z }  \sin  
\frac{\pi t  ^ {2} }{2}
  d t ,
$$

so that

$$ 
C _ {1} ( z)  =  C \left ( \sqrt {
\frac \pi {2}
 } z \right ) \ \ 
\textrm{ and } \ \ 
S _ {1} ( z)  =  S \left ( \sqrt {
\frac \pi {2}
 } z \right ) .
$$

The Fresnel integrals defined in the article are related to the [[Probability integral|probability integral]] for a complex argument  $  z = \sqrt i x $,

$$ 
\Phi ( z)  =  
\frac{2}{\sqrt \pi }

\int\limits _ { 0 } ^ { z }  e ^ {- t  ^ {2} }  d t
$$

(integration along the line  $   \mathop{\rm Re}  z =  \mathop{\rm Im}  z $), 
by

$$ 

\frac{\Phi ( \sqrt i x ) }{\sqrt i }
  = \ 
\sqrt 2 C ( x) - i \sqrt 2 S ( x) .
$$

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Segun,   M. Abramowitz,   "Handbook of mathematical functions" , ''Appl. Math. Ser.'' , '''55''' , Nat. Bur. Standards  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Spanier,   K.B. Oldham,   "An atlas of functions" , Hemisphere &amp; Springer  (1987)  pp. Chapt. 39</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.N. Lebedev,   "Special functions and their applications" , Dover, reprint  (1972)  pp. 21–33  (Translated from Russian)</TD></TR></table>