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The special function defined for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i0514401.png" /> by the equation
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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/i051440/i0514402.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
The special function defined for real  $  x \neq 0 $
 +
by the equation
 +
 
 +
$$
 +
\mathop{\rm Ei} ( x)  = \
 +
\int\limits _ {- \infty } ^ { x }
 +
 
 +
\frac{e  ^ {t} }{t}
 +
  d t  = -
 +
\int\limits _ { - } x ^  \infty 
 +
 
 +
\frac{e  ^ {-} t }{t}
 +
  d t .
 +
$$
  
 
The graph of the integral exponential function is illustrated in Fig..
 
The graph of the integral exponential function is illustrated in Fig..
Line 9: Line 32:
 
Figure: i051440a
 
Figure: i051440a
  
Graphs of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i0514403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i0514404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i0514405.png" />.
+
Graphs of the functions $  y = \mathop{\rm Ei} ( - x ) $,
 +
$  y = \mathop{\rm Ei}  ^ {*} ( x) $
 +
and  $  y = \mathop{\rm Li} ( x) $.
 +
 
 +
For  $  x > 0 $,  
 +
the function  $  e  ^ {t} / t $
 +
has an infinite discontinuity at  $  t = 0 $,
 +
and the integral exponential function is understood in the sense of the principal value of this integral:
 +
 
 +
$$
 +
\mathop{\rm Ei} ( x)  = \
 +
\lim\limits _ {\epsilon \rightarrow + 0 } \
 +
\left \{
 +
\int\limits _ {- \infty } ^  \epsilon 
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i0514406.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i0514407.png" /> has an infinite discontinuity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i0514408.png" />, and the integral exponential function is understood in the sense of the principal value of this integral:
+
\frac{e  ^ {t} }{t}
 +
  d t +
 +
\int\limits _  \epsilon  ^ { 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/i051440/i0514409.png" /></td> </tr></table>
+
\frac{e  ^ {t} }{t}
 +
  d t
 +
\right \} .
 +
$$
  
 
The integral exponential function can be represented by the series
 
The integral exponential function can be represented by 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/i/i051/i051440/i05144010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\mathop{\rm Ei} ( x)  = \
 +
c +  \mathop{\rm ln} ( - x ) +
 +
\sum _ { k= } 1 ^  \infty 
 +
 
 +
\frac{x  ^ {k} }{k!k}
 +
,\ \
 +
x < 0 ,
 +
$$
  
 
and
 
and
  
<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/i051440/i05144011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\mathop{\rm Ei} ( x)  = c +
 +
\mathop{\rm ln} ( x) +
 +
\sum _ { k= } 1 ^  \infty 
 +
 
 +
\frac{x  ^ {k} }{k!k}
 +
,\ \
 +
x > 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144012.png" /> is the [[Euler constant|Euler constant]].
+
where $  c = 0.5772 \dots $
 +
is the [[Euler constant|Euler constant]].
  
 
There is an asymptotic representation:
 
There is an asymptotic 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/i051440/i05144013.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ei} ( - x )  \approx \
 +
 
 +
\frac{e  ^ {-} x }{x}
 +
 
 +
\left (
 +
1 - 1! over {x} +
 +
2! over {x  ^ {2} } -
 +
3! over {x  ^ {3} } + \dots
 +
\right ) ,\ \
 +
x \rightarrow + \infty .
 +
$$
 +
 
 +
As a function of the complex variable  $  z $,
 +
the integral exponential function
 +
 
 +
$$
 +
\mathop{\rm Ei} ( z)  = \
 +
C +  \mathop{\rm ln} ( - z ) +
 +
\sum _ { k= } 1 ^  \infty 
 +
 
 +
\frac{z  ^ {k} }{k!k}
 +
,\ \
 +
|  \mathop{\rm arg} ( - z ) | < \pi ,
 +
$$
  
As a function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144014.png" />, the integral exponential function
+
is a single-valued analytic function in the  $  z $-
 +
plane slit along the positive real semi-axis  $  ( 0 <  \mathop{\rm arg}  z < 2 \pi ) $;
 +
here the value of $  \mathop{\rm ln} ( - z) $
 +
is chosen such that  $  - \pi < { \mathop{\rm Im}  \mathop{\rm ln} } (- z) < \pi $.  
 +
The behaviour of  $  \mathop{\rm Ei} ( z) $
 +
close to the slit is described by the limiting relations:
  
<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/i051440/i05144015.png" /></td> </tr></table>
+
$$
 +
\left .
 +
\begin{array}{c}
 +
\lim\limits _ {\eta \downarrow 0 } \
 +
\mathop{\rm Ei} ( z + i \eta )  = \
 +
\mathop{\rm Ei} ( z) - i \pi ,  \\
 +
\lim\limits _ {\eta \downarrow 0 } \
 +
\mathop{\rm Ei} ( z - i \eta )  = \
 +
\mathop{\rm Ei} ( z) + i \pi ,  \\
 +
\end{array}
 +
\right \} \ \
 +
z = x + i y.
 +
$$
  
is a single-valued analytic function in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144016.png" />-plane slit along the positive real semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144017.png" />; here the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144018.png" /> is chosen such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144019.png" />. The behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144020.png" /> close to the slit is described by the limiting relations:
+
The asymptotic representation in the region  $  0 < \mathop{\rm arg}  z < 2 \pi $
 +
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/i/i051/i051440/i05144021.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ei} ( z)  \sim \
  
The asymptotic representation in the region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144022.png" /> is:
+
\frac{e  ^ {z} }{z}
  
<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/i051440/i05144023.png" /></td> </tr></table>
+
\left (
 +
1! over {z} +
 +
2! over {z  ^ {2} } + \dots
 +
+ k! over {z  ^ {k} } + \dots
 +
\right ) ,\ \
 +
| z | \rightarrow \infty .
 +
$$
  
The integral exponential function is related to the [[Integral logarithm|integral logarithm]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144024.png" /> by the formulas
+
The integral exponential function is related to the [[Integral logarithm|integral logarithm]] $  \mathop{\rm li} ( x) $
 +
by the formulas
  
<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/i051440/i05144025.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ei} ( x)  = \
 +
\mathop{\rm li} ( e  ^ {x} ) ,\ \
 +
x < 0 ,
 +
$$
  
<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/i051440/i05144026.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ei} (  \mathop{\rm ln}  x )  =   \mathop{\rm li} ( x) ,\  x < 1 ;
 +
$$
  
and to the [[Integral sine|integral sine]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144027.png" /> and the [[Integral cosine|integral cosine]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144028.png" /> by the formulas:
+
and to the [[Integral sine|integral sine]] $  \mathop{\rm Si} ( x) $
 +
and the [[Integral cosine|integral cosine]] $  \mathop{\rm Ci} ( x) $
 +
by the formulas:
  
<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/i051440/i05144029.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ei} ( \pm  i x )  = \
 +
\mathop{\rm Ci} ( x) \pm  i  \mathop{\rm Si} ( x) \mps
 +
\frac{\pi i }{2}
 +
,\ \
 +
x > 0 .
 +
$$
  
 
The differentiation formula is:
 
The differentiation formula 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/i/i051/i051440/i05144030.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{d  ^ {n}  \mathop{\rm Ei} ( - x ) }{d x  ^ {n} }
 +
  = \
 +
( - 1 )  ^ {n-} 1
 +
( n - 1 ) ! x  ^ {-} x
 +
e  ^ {-} x e _ {n-} 1 ( x) ,\ \
 +
n = 1 , 2 , .  . . .
 +
$$
  
 
The following notations are sometimes used:
 
The following notations are sometimes used:
  
<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/i051440/i05144031.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ei}  ^ {+} ( z)  = \
 +
\mathop{\rm Ei} ( z + i 0 ) ,\ \
 +
\mathop{\rm Ei}  ^ {-} ( z)  = \
 +
\mathop{\rm Ei} ( z - i 0 ) ,
 +
$$
  
<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/i051440/i05144032.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ei}  ^ {*} ( z)  = { \mathop{\rm Ei} ( z) } bar  =   \mathop{\rm Ei} ( z) + \pi i .
 +
$$
  
 
====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. Krazer,  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. Krazer,  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/i051440/i05144033.png" /> is usually called the exponential integral.
+
The function $  \mathop{\rm Ei} $
 +
is usually called the exponential integral.
  
Instead of by the series representation, for complex values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144035.png" /> not positive real) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144036.png" /> can be defined by the integal (as for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144037.png" />); since the integrand is analytic, the integral is path-independent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144038.png" />.
+
Instead of by the series representation, for complex values of $  z $(
 +
$  x $
 +
not positive real) the function $  \mathop{\rm Ei} ( z) $
 +
can be defined by the integal (as for real $  x \neq 0 $);  
 +
since the integrand is analytic, the integral is path-independent in $  \mathbf C \setminus  \{ {x \in \mathbf R } : {x \geq  0 } \} $.
  
Formula (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144039.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144040.png" /> then holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144041.png" />, and the function defined by (2) (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051440/i05144042.png" />) is also known as the modified exponential integral.
+
Formula (1) with $  x $
 +
replaced by $  z $
 +
then holds for $  |  \mathop{\rm arg} ( - z ) | < \pi $,  
 +
and the function defined by (2) (for $  x > 0 $)  
 +
is also known as the modified exponential integral.

Revision as of 22:12, 5 June 2020


The special function defined for real $ x \neq 0 $ by the equation

$$ \mathop{\rm Ei} ( x) = \ \int\limits _ {- \infty } ^ { x } \frac{e ^ {t} }{t} d t = - \int\limits _ { - } x ^ \infty \frac{e ^ {-} t }{t} d t . $$

The graph of the integral exponential function is illustrated in Fig..

Figure: i051440a

Graphs of the functions $ y = \mathop{\rm Ei} ( - x ) $, $ y = \mathop{\rm Ei} ^ {*} ( x) $ and $ y = \mathop{\rm Li} ( x) $.

For $ x > 0 $, the function $ e ^ {t} / t $ has an infinite discontinuity at $ t = 0 $, and the integral exponential function is understood in the sense of the principal value of this integral:

$$ \mathop{\rm Ei} ( x) = \ \lim\limits _ {\epsilon \rightarrow + 0 } \ \left \{ \int\limits _ {- \infty } ^ \epsilon \frac{e ^ {t} }{t} d t + \int\limits _ \epsilon ^ { x } \frac{e ^ {t} }{t} d t \right \} . $$

The integral exponential function can be represented by the series

$$ \tag{1 } \mathop{\rm Ei} ( x) = \ c + \mathop{\rm ln} ( - x ) + \sum _ { k= } 1 ^ \infty \frac{x ^ {k} }{k!k} ,\ \ x < 0 , $$

and

$$ \tag{2 } \mathop{\rm Ei} ( x) = c + \mathop{\rm ln} ( x) + \sum _ { k= } 1 ^ \infty \frac{x ^ {k} }{k!k} ,\ \ x > 0 , $$

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

There is an asymptotic representation:

$$ \mathop{\rm Ei} ( - x ) \approx \ \frac{e ^ {-} x }{x} \left ( 1 - 1! over {x} + 2! over {x ^ {2} } - 3! over {x ^ {3} } + \dots \right ) ,\ \ x \rightarrow + \infty . $$

As a function of the complex variable $ z $, the integral exponential function

$$ \mathop{\rm Ei} ( z) = \ C + \mathop{\rm ln} ( - z ) + \sum _ { k= } 1 ^ \infty \frac{z ^ {k} }{k!k} ,\ \ | \mathop{\rm arg} ( - z ) | < \pi , $$

is a single-valued analytic function in the $ z $- plane slit along the positive real semi-axis $ ( 0 < \mathop{\rm arg} z < 2 \pi ) $; here the value of $ \mathop{\rm ln} ( - z) $ is chosen such that $ - \pi < { \mathop{\rm Im} \mathop{\rm ln} } (- z) < \pi $. The behaviour of $ \mathop{\rm Ei} ( z) $ close to the slit is described by the limiting relations:

$$ \left . \begin{array}{c} \lim\limits _ {\eta \downarrow 0 } \ \mathop{\rm Ei} ( z + i \eta ) = \ \mathop{\rm Ei} ( z) - i \pi , \\ \lim\limits _ {\eta \downarrow 0 } \ \mathop{\rm Ei} ( z - i \eta ) = \ \mathop{\rm Ei} ( z) + i \pi , \\ \end{array} \right \} \ \ z = x + i y. $$

The asymptotic representation in the region $ 0 < \mathop{\rm arg} z < 2 \pi $ is:

$$ \mathop{\rm Ei} ( z) \sim \ \frac{e ^ {z} }{z} \left ( 1! over {z} + 2! over {z ^ {2} } + \dots + k! over {z ^ {k} } + \dots \right ) ,\ \ | z | \rightarrow \infty . $$

The integral exponential function is related to the integral logarithm $ \mathop{\rm li} ( x) $ by the formulas

$$ \mathop{\rm Ei} ( x) = \ \mathop{\rm li} ( e ^ {x} ) ,\ \ x < 0 , $$

$$ \mathop{\rm Ei} ( \mathop{\rm ln} x ) = \mathop{\rm li} ( x) ,\ x < 1 ; $$

and to the integral sine $ \mathop{\rm Si} ( x) $ and the integral cosine $ \mathop{\rm Ci} ( x) $ by the formulas:

$$ \mathop{\rm Ei} ( \pm i x ) = \ \mathop{\rm Ci} ( x) \pm i \mathop{\rm Si} ( x) \mps \frac{\pi i }{2} ,\ \ x > 0 . $$

The differentiation formula is:

$$ \frac{d ^ {n} \mathop{\rm Ei} ( - x ) }{d x ^ {n} } = \ ( - 1 ) ^ {n-} 1 ( n - 1 ) ! x ^ {-} x e ^ {-} x e _ {n-} 1 ( x) ,\ \ n = 1 , 2 , . . . . $$

The following notations are sometimes used:

$$ \mathop{\rm Ei} ^ {+} ( z) = \ \mathop{\rm Ei} ( z + i 0 ) ,\ \ \mathop{\rm Ei} ^ {-} ( z) = \ \mathop{\rm Ei} ( z - i 0 ) , $$

$$ \mathop{\rm Ei} ^ {*} ( z) = { \mathop{\rm Ei} ( z) } bar = \mathop{\rm Ei} ( z) + \pi i . $$

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. Krazer, 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 Ei} $ is usually called the exponential integral.

Instead of by the series representation, for complex values of $ z $( $ x $ not positive real) the function $ \mathop{\rm Ei} ( z) $ can be defined by the integal (as for real $ x \neq 0 $); since the integrand is analytic, the integral is path-independent in $ \mathbf C \setminus \{ {x \in \mathbf R } : {x \geq 0 } \} $.

Formula (1) with $ x $ replaced by $ z $ then holds for $ | \mathop{\rm arg} ( - z ) | < \pi $, and the function defined by (2) (for $ x > 0 $) is also known as the modified exponential integral.

How to Cite This Entry:
Integral exponential function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_exponential_function&oldid=47371
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article