Namespaces
Variants
Actions

Difference between revisions of "Fresnel integrals"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
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
 
The special functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417201.png" /></td> </tr></table>
+
$$
 +
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,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417202.png" /></td> </tr></table>
+
$$
 +
S ( x)  = {
 +
\frac{1}{\sqrt {2 \pi } }
 +
} \int\limits _ { 0 } ^ { {x }
 +
^ {2} }
 +
\frac{\sin  t }{\sqrt t }
 +
  dt  = \sqrt {
 +
\frac{2} \pi
 +
} \int\limits _ { 0 } ^ { x }  \sin  t  ^ {2}  dt;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417203.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow + \infty }  C ( x)  = \lim\limits _ {x \rightarrow + \infty }  S ( x)  = {
 +
\frac{1}{2}
 +
} .
 +
$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041720a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041720a.gif" />
Line 13: Line 51:
 
The Fresnel integrals can be represented in the form of the series
 
The Fresnel integrals can be represented in the form of the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417204.png" /></td> </tr></table>
+
$$
 +
C ( x)  = \sqrt {
 +
\frac{2} \pi
 +
} x
 +
\sum _ {k = 0 } ^  \infty 
 +
 
 +
\frac{(- 1)  ^ {k} x  ^ {4k} }{( 2k)! ( 4k + 1) }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417205.png" /></td> </tr></table>
+
$$
 +
S ( x)  = \sqrt {
 +
\frac{2} \pi
 +
} x \sum _ {k = 0 } ^  \infty 
 +
\frac{(- 1)  ^ {k} x ^ {2 ( 2 k + 1) } }{( 2k + 1)! ( 4k + 3) }
 +
.
 +
$$
  
An asymptotic representation for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417206.png" /> is:
+
An asymptotic representation for large $  x $
 +
is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417207.png" /></td> </tr></table>
+
$$
 +
C ( x)  = {
 +
\frac{1}{2}
 +
} +
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417208.png" /></td> </tr></table>
+
\frac{1}{\sqrt {2 \pi } x }
 +
\
 +
\sin  x  ^ {2} + O
 +
\left (
 +
\frac{1}{x  ^ {2} }
 +
\right ) ,
 +
$$
  
In a rectangular coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f0417209.png" /> the projections of the curve
+
$$
 +
S ( x)  = {
 +
\frac{1}{2}
 +
} -
 +
\frac{1}{\sqrt {2 \pi } x }
 +
  \cos
 +
x  ^ {2} + O \left (
 +
\frac{1}{x  ^ {2} }
 +
\right ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172010.png" /></td> </tr></table>
+
In a rectangular coordinate system  $  ( x, y) $
 +
the projections of the curve
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172011.png" /> is a real parameter, onto the coordinate planes are the [[Cornu spiral|Cornu spiral]] and the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172013.png" /> (see Fig. b). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172014.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172015.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172016.png" />
+
$$
 +
= t,\ \
 +
= C \left ( {
 +
\frac \pi {2}
 +
} t  ^ {2} \right ) ,\ \
 +
= S \left ( {
 +
\frac \pi {2}
 +
} t  ^ {2} \right ) ,
 +
$$
 +
 
 +
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041720b.gif" />
Line 35: Line 122:
 
The generalized Fresnel integrals (see [[#References|[1]]]) are functions of the form
 
The generalized Fresnel integrals (see [[#References|[1]]]) are functions of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172017.png" /></td> </tr></table>
+
$$
 +
C ( x, \alpha )  = \
 +
\int\limits _ { x } ^  \infty 
 +
t ^ {\alpha - 1 } \
 +
\cos  t  dt,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172018.png" /></td> </tr></table>
+
$$
 +
S ( x, \alpha )  = \int\limits _ { x } ^  \infty  t ^ {\alpha - 1 }  \sin  t  dt.
 +
$$
  
 
The Fresnel integrals are related to the generalized Fresnel integrals as follows:
 
The Fresnel integrals are related to the generalized Fresnel integrals as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172019.png" /></td> </tr></table>
+
$$
 +
C ( x)  = {
 +
\frac{1}{2}
 +
} -
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172020.png" /></td> </tr></table>
+
\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====
 
====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>
 
<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====
 
====Comments====
 
A word of warning. There are different normalizations in use for the Fresnel integrals. E.g., in [[#References|[a3]]] they are defined as
 
A word of warning. There are different normalizations in use for the Fresnel integrals. E.g., in [[#References|[a3]]] they are defined as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172021.png" /></td> </tr></table>
+
$$
 +
C _ {1} ( z)  = \int\limits _ { 0 } ^ { z }
 +
\cos 
 +
\frac{\pi t  ^ {2} }{2}
 +
  d t ,\  S _ {1} ( z)  = \
 +
\int\limits _ { 0 } ^ { z }  \sin 
 +
\frac{\pi t  ^ {2} }{2}
 +
  d t ,
 +
$$
  
 
so that
 
so that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172022.png" /></td> </tr></table>
+
$$
 +
C _ {1} ( z)  = C \left ( \sqrt {
 +
\frac \pi {2}
 +
} z \right ) \ \
 +
\textrm{ and } \ \
 +
S _ {1} ( z)  = S \left ( \sqrt {
 +
\frac \pi {2}
 +
} z \right ) .
 +
$$
 +
 
 +
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 }
  
The Fresnel integrals defined in the article are related to the [[Probability integral|probability integral]] for a complex argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172023.png" />,
+
\int\limits _ { 0 } ^ { z }  e ^ {- t  ^ {2} }  d t
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172024.png" /></td> </tr></table>
+
(integration along the line  $  \mathop{\rm Re}  z = \mathop{\rm Im}  z $),
 +
by
  
(integration along the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172025.png" />), by
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041720/f04172026.png" /></td> </tr></table>
+
\frac{\Phi ( \sqrt i x ) }{\sqrt i }
 +
  = \
 +
\sqrt 2 C ( x) - i \sqrt 2 S ( x) .
 +
$$
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:40, 5 June 2020


The special functions

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

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

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

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 and the curves $ y = C ( \pi t ^ {2} /2) $, $ z = S ( \pi t ^ {2} /2) $( see Fig. b). $ x $ $ S( x) $ $ C( x) $

Figure: f041720b

The generalized Fresnel integrals (see [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

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

Comments

A word of warning. There are different normalizations in use for the Fresnel integrals. E.g., in [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 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

[a1] A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)
[a2] J. Spanier, K.B. Oldham, "An atlas of functions" , Hemisphere & Springer (1987) pp. Chapt. 39
[a3] N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) pp. 21–33 (Translated from Russian)
How to Cite This Entry:
Fresnel integrals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fresnel_integrals&oldid=46990
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article