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The special function defined, for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i0513701.png" />, by
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<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/i/i051/i051370/i0513702.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i0513703.png" /> is the [[Euler constant|Euler constant]]. Its graph is:
+
The special function defined, for real  $  x > 0 $,
 +
by
 +
 
 +
$$
 +
\mathop{\rm Ci} ( x)  = -
 +
\int\limits _ { x } ^  \infty 
 +
 
 +
\frac{\cos  t }{t }
 +
\
 +
d t  = c +  \mathop{\rm ln}  x -
 +
\int\limits _ { 0 } ^ { x }
 +
 
 +
\frac{1 - \cos  t }{t }
 +
\
 +
d t ,
 +
$$
 +
 
 +
where  $  c = 0.5772 \dots $
 +
is the [[Euler constant|Euler constant]]. Its graph is:
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i051370a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i051370a.gif" />
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Figure: i051370a
 
Figure: i051370a
  
The graphs of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i0513704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i0513705.png" />.
+
The graphs of the functions $  y = \mathop{\rm ci} ( x) $
 +
and $  y = \mathop{\rm si} ( x) $.
  
 
Some integrals related to the integral cosine are:
 
Some integrals related to the integral cosine are:
  
<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/i/i051/i051370/i0513706.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
e ^ {- p t }  \mathop{\rm Ci} ( q t )  d t  = -
 +
 
 +
\frac{1}{2p}
 +
  \mathop{\rm ln}
 +
\left (
 +
1 +
 +
\frac{p  ^ {2} }{q  ^ {2} }
 +
 
 +
\right ) ,
 +
$$
 +
 
 +
$$
 +
\int\limits _ { 0 } ^  \infty  \cos  t  \mathop{\rm Ci} ( t) \
 +
d t  = -  
 +
\frac \pi {4}
 +
,\  \int\limits _ { 0 } ^  \infty    \mathop{\rm Ci}  ^ {2} ( t)  d t  =
 +
\frac \pi {2}
 +
,
 +
$$
  
<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/i/i051/i051370/i0513707.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty    \mathop{\rm Ci} ( t)  \mathop{\rm si} ( t)  d t  = - \mathop{\rm ln}  2 ,
 +
$$
  
<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/i/i051/i051370/i0513708.png" /></td> </tr></table>
+
where  $  \mathop{\rm si} ( t) $
 +
is the [[Integral sine|integral sine]] minus  $  \pi / 2 $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i0513709.png" /> is the [[Integral sine|integral sine]] minus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137010.png" />.
+
For  $  x $
 +
small:
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137011.png" /> small:
+
$$
 +
\mathop{\rm Ci} ( x)  \approx  c +  \mathop{\rm ln}  x .
 +
$$
  
<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/i/i051/i051370/i05137012.png" /></td> </tr></table>
+
The asymptotic representation, for  $  x $
 +
large, is:
  
The asymptotic representation, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137013.png" /> large, is:
+
$$
 +
\mathop{\rm Ci} ( x)  = \
  
<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/i/i051/i051370/i05137014.png" /></td> </tr></table>
+
\frac{\sin  x }{x}
 +
P ( x) -
  
<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/i/i051/i051370/i05137015.png" /></td> </tr></table>
+
\frac{\cos  x }{x}
 +
Q ( x) ,
 +
$$
 +
 
 +
$$
 +
P ( x)  \sim  \sum _ { k= } 0 ^  \infty 
 +
\frac{( - 1 )  ^ {k} (
 +
2 k ) ! }{x  ^ {2k} }
 +
,\  Q ( x)  \sim  \sum _ { k= } 0 ^  \infty 
 +
\frac{( - 1 )  ^ {k} ( 2 k + 1 ) ! }{x  ^ {2k+} 1 }
 +
.
 +
$$
  
 
The integral cosine has the series representation:
 
The integral cosine has the series representation:
  
<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/i/i051/i051370/i05137016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\mathop{\rm Ci} ( x)  = c +  \mathop{\rm ln}  x -
 +
\frac{x  ^ {2} }{2!2}
 +
 
 +
+
 +
\frac{x  ^ {4} }{4!4}
 +
- \dots +
 +
$$
 +
 
 +
$$
 +
+
 +
( - 1 )  ^ {k}
 +
\frac{x  ^ {2k} }{( 2 k ) ! 2 k }
 +
+ \dots .
 +
$$
  
<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/i/i051/i051370/i05137017.png" /></td> </tr></table>
+
As a function of the complex variable  $  z $,
 +
$  \mathop{\rm Ci} ( z) $,
 +
defined by (*), is a single-valued analytic function in the  $  z $-
 +
plane with slit along the relative negative real axis  $  ( - \pi < \mathop{\rm arg}  z < \pi ) $.
 +
The value of  $  \mathop{\rm ln}  z $
 +
here is taken to be  $  \pi < \mathop{\rm Im}  \mathop{\rm ln}  z < \pi $.  
 +
The behaviour of  $  \mathop{\rm Ci} ( z) $
 +
near the slit is determined by the limits
  
As a function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137019.png" />, defined by (*), is a single-valued analytic function in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137020.png" />-plane with slit along the relative negative real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137021.png" />. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137022.png" /> here is taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137023.png" />. The behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137024.png" /> near the slit is determined by the limits
+
$$
 +
\lim\limits _ {\eta \downarrow 0 }  \mathop{\rm Ci} ( x \pm  i \eta )  = \
 +
\mathop{\rm Ci} ( | z | ) \pm  \pi i ,\  x < 0 .
 +
$$
  
<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/i/i051/i051370/i05137025.png" /></td> </tr></table>
+
The integral cosine is related to the [[Integral exponential function|integral exponential function]]  $  \mathop{\rm Ei} ( z) $
 +
by
  
The integral cosine is related to the [[Integral exponential function|integral exponential function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137026.png" /> by
+
$$
 +
\mathop{\rm Ci} ( z)  =
 +
\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/i/i051/i051370/i05137027.png" /></td> </tr></table>
+
[  \mathop{\rm Ei} ( i z ) +  \mathop{\rm Ei} ( - i z ) ] .
 +
$$
  
One sometimes uses the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137028.png" />.
+
One sometimes uses the notation $  \mathop{\rm ci} ( x) \equiv  \mathop{\rm Ci} ( x) $.
  
 
See also [[Si-ci-spiral|Si-ci-spiral]].
 
See also [[Si-ci-spiral|Si-ci-spiral]].
Line 51: Line 146:
 
====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><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Kratzer,  W. Franz,  "Transzendente Funktionen" , Akademie Verlag  (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.N. Lebedev,  "Special functions and their applications" , Prentice-Hall  (1965)  (Translated from Russian)</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><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Kratzer,  W. Franz,  "Transzendente Funktionen" , Akademie Verlag  (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.N. Lebedev,  "Special functions and their applications" , Prentice-Hall  (1965)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137029.png" /> is better known as the cosine integral. It can, of course, be defined by the integral (as above) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051370/i05137030.png" />.
+
The function $  \mathop{\rm Ci} $
 +
is better known as the cosine integral. It can, of course, be defined by the integral (as above) in $  \mathbf C \setminus  \{ {x \in \mathbf R } : {x \leq  0 } \} $.

Revision as of 22:12, 5 June 2020


The special function defined, for real $ x > 0 $, by

$$ \mathop{\rm Ci} ( x) = - \int\limits _ { x } ^ \infty \frac{\cos t }{t } \ d t = c + \mathop{\rm ln} x - \int\limits _ { 0 } ^ { x } \frac{1 - \cos t }{t } \ d t , $$

where $ c = 0.5772 \dots $ is the Euler constant. Its graph is:

Figure: i051370a

The graphs of the functions $ y = \mathop{\rm ci} ( x) $ and $ y = \mathop{\rm si} ( x) $.

Some integrals related to the integral cosine are:

$$ \int\limits _ { 0 } ^ \infty e ^ {- p t } \mathop{\rm Ci} ( q t ) d t = - \frac{1}{2p} \mathop{\rm ln} \left ( 1 + \frac{p ^ {2} }{q ^ {2} } \right ) , $$

$$ \int\limits _ { 0 } ^ \infty \cos t \mathop{\rm Ci} ( t) \ d t = - \frac \pi {4} ,\ \int\limits _ { 0 } ^ \infty \mathop{\rm Ci} ^ {2} ( t) d t = \frac \pi {2} , $$

$$ \int\limits _ { 0 } ^ \infty \mathop{\rm Ci} ( t) \mathop{\rm si} ( t) d t = - \mathop{\rm ln} 2 , $$

where $ \mathop{\rm si} ( t) $ is the integral sine minus $ \pi / 2 $.

For $ x $ small:

$$ \mathop{\rm Ci} ( x) \approx c + \mathop{\rm ln} x . $$

The asymptotic representation, for $ x $ large, is:

$$ \mathop{\rm Ci} ( x) = \ \frac{\sin x }{x} P ( x) - \frac{\cos x }{x} Q ( x) , $$

$$ P ( x) \sim \sum _ { k= } 0 ^ \infty \frac{( - 1 ) ^ {k} ( 2 k ) ! }{x ^ {2k} } ,\ Q ( x) \sim \sum _ { k= } 0 ^ \infty \frac{( - 1 ) ^ {k} ( 2 k + 1 ) ! }{x ^ {2k+} 1 } . $$

The integral cosine has the series representation:

$$ \tag{* } \mathop{\rm Ci} ( x) = c + \mathop{\rm ln} x - \frac{x ^ {2} }{2!2} + \frac{x ^ {4} }{4!4} - \dots + $$

$$ + ( - 1 ) ^ {k} \frac{x ^ {2k} }{( 2 k ) ! 2 k } + \dots . $$

As a function of the complex variable $ z $, $ \mathop{\rm Ci} ( z) $, defined by (*), is a single-valued analytic function in the $ z $- plane with slit along the relative negative real axis $ ( - \pi < \mathop{\rm arg} z < \pi ) $. The value of $ \mathop{\rm ln} z $ here is taken to be $ \pi < \mathop{\rm Im} \mathop{\rm ln} z < \pi $. The behaviour of $ \mathop{\rm Ci} ( z) $ near the slit is determined by the limits

$$ \lim\limits _ {\eta \downarrow 0 } \mathop{\rm Ci} ( x \pm i \eta ) = \ \mathop{\rm Ci} ( | z | ) \pm \pi i ,\ x < 0 . $$

The integral cosine is related to the integral exponential function $ \mathop{\rm Ei} ( z) $ by

$$ \mathop{\rm Ci} ( z) = \frac{1}{2} [ \mathop{\rm Ei} ( i z ) + \mathop{\rm Ei} ( - i z ) ] . $$

One sometimes uses the notation $ \mathop{\rm ci} ( x) \equiv \mathop{\rm Ci} ( x) $.

See also Si-ci-spiral.

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)
[3] A. Kratzer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)
[4] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)

Comments

The function $ \mathop{\rm Ci} $ is better known as the cosine integral. It can, of course, be defined by the integral (as above) in $ \mathbf C \setminus \{ {x \in \mathbf R } : {x \leq 0 } \} $.

How to Cite This Entry:
Integral cosine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_cosine&oldid=47367
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article