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A summation formula that connects the partial sums of a series with the integral and derivatives of its general term:
 
A summation formula that connects the partial sums of a series with the integral and derivatives of its general term:
  
<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/e/e036/e036520/e0365201.png" /></td> </tr></table>
+
$$
 +
\sum _ {k=p}^ {m-1} \phi ( k)  = \int\limits _ { p } ^ { m }
 +
\phi ( t)  dt +
 +
$$
  
<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/e/e036/e036520/e0365202.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ {\nu = 1 } ^ {n-1}
 +
\frac{B _  \nu  }{
 +
\nu ! }
 +
\{ \phi ^ {( \nu - 1 ) } ( m) -
 +
\phi ^ {( \nu - 1 ) } ( p) \} + R _ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365203.png" /> are the [[Bernoulli numbers|Bernoulli numbers]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365204.png" /> is the remainder. Using the [[Bernoulli polynomials|Bernoulli polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365206.png" />, the remainder can be rewritten in the form
+
where $  B _  \nu  $
 +
are the [[Bernoulli numbers]] and $  R _ {n} $
 +
is the remainder. Using the [[Bernoulli polynomials]] $  b _ {n} ( t) $,  
 +
$  b _ {n} ( 0) = B _ {n} $,  
 +
the remainder can be rewritten in 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/e/e036/e036520/e0365207.png" /></td> </tr></table>
+
$$
 +
R _ {n}  = -  
 +
\frac{1}{n!}
 +
\int\limits _ { 0 } ^ { 1 }
 +
[ B _ {n} ( t) - B _ {n} ] \sum _ {k=p}^ {m-1} \phi  ^ {(} n)
 +
( k + 1 - t )  dt .
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365208.png" /> the remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365209.png" /> can be expressed by means of the Bernoulli numbers:
+
For $  n = 2s $
 +
the remainder $  R _ {2s} $
 +
can be expressed by means of the Bernoulli numbers:
  
<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/e/e036/e036520/e03652010.png" /></td> </tr></table>
+
$$
 +
R _ {2s}  =
 +
\frac{B _ {2s} }{( 2 s ) ! }
  
If the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652012.png" /> have the same sign and do not change sign on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652013.png" />, then
+
\sum _ { k=p} ^ { m-1} \phi  ^ {( 2s) }( k + \theta ) ,\ \
 +
0 < \theta < 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/e/e036/e036520/e03652014.png" /></td> </tr></table>
+
If the derivatives  $  \phi  ^ {( 2s)} ( t) $
 +
and  $  \phi  ^ {( 2s+ 1)} ( t) $
 +
have the same sign and do not change sign on  $  [ p , m ] $,
 +
then
 +
 
 +
$$
 +
R _ {2s}  = \theta
 +
\frac{B _ {2s} }{( 2 s ) ! }
 +
[ \phi  ^ {( 2s- 1)} ( m) - \phi ^ {( 2s- 1) } ( p) ] ,\  0 \leq  \theta \leq  1 .
 +
$$
  
 
If, furthermore,
 
If, furthermore,
  
<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/e/e036/e036520/e03652015.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow \infty }  \phi  ^ {( 2s- 1)} ( x)  = 0 ,
 +
$$
  
 
then the Euler–MacLaurin formula becomes
 
then the Euler–MacLaurin formula becomes
  
<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/e/e036/e036520/e03652016.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= p} ^ { m-1} \phi ( k)  = c +
 +
\int\limits _ { p } ^ { m }  \phi ( t)  dt +
 +
$$
 +
 
 +
$$
 +
+
 +
\sum _ { k= 1} ^ { 2s-  2 }
 +
\frac{B _ {k} }{k ! }
 +
\phi ^ {( k - 1 ) } ( m)
 +
+ \theta
 +
\frac{B _ {2s} }{( 2 s ) ! }
 +
\phi ^ {( 2 s - 1 ) } ( m) ,\  0 < \theta < 1 .
 +
$$
 +
 
 +
This version is used, for example, to derive the [[Stirling formula]], in which case  $  \phi ( x) = \mathop{\rm ln}  x $
 +
and  $  c $
 +
is the [[Euler constant]]. The formula has also been generalized to multiple sums.
  
<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/e/e036/e036520/e03652017.png" /></td> </tr></table>
+
The Euler–MacLaurin formula finds application in the approximate calculation of definite integrals, the study of convergence of series, the computation of sums, and the expansion of functions in Taylor series. For example, for  $  m = 1 $,
 +
$  p = 0 $,
 +
$  n = 2m + 1 $,
 +
and  $  \phi ( x) = \cos ( x t - t / 2 ) $,
 +
it yields the expression
  
This version is used, for example, to derive the [[Stirling formula|Stirling formula]], in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652019.png" /> is the [[Euler constant|Euler constant]]. The formula has also been generalized to multiple sums.
+
$$
  
The Euler–MacLaurin formula finds application in the approximate calculation of definite integrals, the study of convergence of series, the computation of sums, and the expansion of functions in Taylor series. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652023.png" />, it yields the expression
+
\frac{t}{2}
 +
  \mathop{\rm cotan} 
 +
\frac{t}{2}
 +
  = \sum _ {\nu = 0 } ^ { m }
 +
(- 1)  ^  \nu 
 +
\frac{t ^ {2 \nu } }{( 2 \nu ) ! }
 +
B _ {2 \nu }  +
 +
$$
  
<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/e/e036/e036520/e03652024.png" /></td> </tr></table>
+
$$
 +
+
  
<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/e/e036/e036520/e03652025.png" /></td> </tr></table>
+
\frac{( - 1 ) ^ {m + 1 } t  ^ {2m+ 2} }{2 \
 +
\sin ( t / 2) }
 +
\int\limits _ { 0 } ^ { 1 } 
 +
\frac{B _ {2m+ 1} ( t) }{( 2
 +
m + 1 ) ! }
 +
\sin \left ( x -
 +
\frac{1}{2}
 +
\right ) t  dx .
 +
$$
  
 
The Euler–MacLaurin formula plays an important role in the study of asymptotic expansions, number-theoretic estimates and [[Finite-difference calculus|finite-difference calculus]].
 
The Euler–MacLaurin formula plays an important role in the study of asymptotic expansions, number-theoretic estimates and [[Finite-difference calculus|finite-difference calculus]].
Line 39: Line 125:
 
Sometimes the Euler–MacLaurin formula is applied in the form
 
Sometimes the Euler–MacLaurin formula is applied in 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/e/e036/e036520/e03652026.png" /></td> </tr></table>
+
$$
 +
\sum _ { 0 } ^ { n }  \phi _ {n} ( x)  = \int\limits _ { 0 } ^ { n }  \phi ( x) \
 +
dx +
 +
\frac{1}{2}
 +
( \phi _ {0} + \phi _ {n} ) +
 +
$$
  
<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/e/e036/e036520/e03652027.png" /></td> </tr></table>
+
$$
 +
+
 +
\int\limits _ { 0 } ^ { n }  \left ( x - [ x] -
 +
\frac{1}{2}
 +
\right ) \phi  ^  \prime  ( x)  dx .
 +
$$
  
 
The formula was first obtained by L. Euler [[#References|[1]]] as
 
The formula was first obtained by L. Euler [[#References|[1]]] 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/e/e036/e036520/e03652028.png" /></td> </tr></table>
+
$$
 +
= \int\limits t  dn + \alpha t + \beta 
 +
\frac{dt}{dn}
 +
+
 +
\gamma
 +
\frac{d  ^ {2} t }{d n  ^ {2} }
 +
+ \delta
 +
 
 +
\frac{d  ^ {2} t }{d n  ^ {2} }
 +
+
 +
\epsilon
 +
\frac{d  ^ {4} t }{d n  ^ {4} }
 +
+ \dots ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652029.png" /> is the sum of the first terms of the series with general term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652032.png" />, and the coefficients are determined from the recurrence relations
+
where $  S $
 +
is the sum of the first terms of the series with general term $  t ( n) $,  
 +
$  S = t = 0 $
 +
for $  n = 0 $,  
 +
and the coefficients are determined from the recurrence 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/e/e036/e036520/e03652033.png" /></td> </tr></table>
+
$$
 +
\alpha  =
 +
\frac{1}{2}
 +
,\  \beta  =
 +
\frac \alpha {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/e/e036/e036520/e03652034.png" /></td> </tr></table>
+
\frac{1}{3!}
 +
  =
 +
\frac{1}{12}
 +
,\  \gamma  =
 +
\frac \beta {2!}
 +
-
 +
 
 +
\frac \alpha {3!}
 +
+
 +
\frac{1}{4!}
 +
  = 0 ,
 +
$$
 +
 
 +
$$
 +
\delta  =
 +
\frac \gamma {2!}
 +
-
 +
\frac \beta {3!}
 +
+
 +
\frac \alpha {4!}
 +
-  
 +
\frac{1}{5!}
 +
  = -
 +
\frac{1}{720}
 +
,\  \epsilon  = 0 ,\  \gamma  = 0 ,\dots .
 +
$$
  
 
The formula was later discovered independently by C. MacLaurin [[#References|[2]]].
 
The formula was later discovered independently by C. MacLaurin [[#References|[2]]].
Line 57: Line 200:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  ''Comment. Acad. Sci. Imp. Petrop.'' , '''6'''  (1738)  pp. 68–97</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. MacLaurin,  "A treatise of fluxions" , '''1–2''' , Edinburgh  (1742)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.O. [A.O. Gel'fond] Gelfond,  "Differenzenrechnung" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  ''Comment. Acad. Sci. Imp. Petrop.'' , '''6'''  (1738)  pp. 68–97</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. MacLaurin,  "A treatise of fluxions" , '''1–2''' , Edinburgh  (1742)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.O. [A.O. Gel'fond] Gelfond,  "Differenzenrechnung" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 09:18, 6 January 2024


A summation formula that connects the partial sums of a series with the integral and derivatives of its general term:

$$ \sum _ {k=p}^ {m-1} \phi ( k) = \int\limits _ { p } ^ { m } \phi ( t) dt + $$

$$ + \sum _ {\nu = 1 } ^ {n-1} \frac{B _ \nu }{ \nu ! } \{ \phi ^ {( \nu - 1 ) } ( m) - \phi ^ {( \nu - 1 ) } ( p) \} + R _ {n} , $$

where $ B _ \nu $ are the Bernoulli numbers and $ R _ {n} $ is the remainder. Using the Bernoulli polynomials $ b _ {n} ( t) $, $ b _ {n} ( 0) = B _ {n} $, the remainder can be rewritten in the form

$$ R _ {n} = - \frac{1}{n!} \int\limits _ { 0 } ^ { 1 } [ B _ {n} ( t) - B _ {n} ] \sum _ {k=p}^ {m-1} \phi ^ {(} n) ( k + 1 - t ) dt . $$

For $ n = 2s $ the remainder $ R _ {2s} $ can be expressed by means of the Bernoulli numbers:

$$ R _ {2s} = \frac{B _ {2s} }{( 2 s ) ! } \sum _ { k=p} ^ { m-1} \phi ^ {( 2s) }( k + \theta ) ,\ \ 0 < \theta < 1 . $$

If the derivatives $ \phi ^ {( 2s)} ( t) $ and $ \phi ^ {( 2s+ 1)} ( t) $ have the same sign and do not change sign on $ [ p , m ] $, then

$$ R _ {2s} = \theta \frac{B _ {2s} }{( 2 s ) ! } [ \phi ^ {( 2s- 1)} ( m) - \phi ^ {( 2s- 1) } ( p) ] ,\ 0 \leq \theta \leq 1 . $$

If, furthermore,

$$ \lim\limits _ {x \rightarrow \infty } \phi ^ {( 2s- 1)} ( x) = 0 , $$

then the Euler–MacLaurin formula becomes

$$ \sum _ { k= p} ^ { m-1} \phi ( k) = c + \int\limits _ { p } ^ { m } \phi ( t) dt + $$

$$ + \sum _ { k= 1} ^ { 2s- 2 } \frac{B _ {k} }{k ! } \phi ^ {( k - 1 ) } ( m) + \theta \frac{B _ {2s} }{( 2 s ) ! } \phi ^ {( 2 s - 1 ) } ( m) ,\ 0 < \theta < 1 . $$

This version is used, for example, to derive the Stirling formula, in which case $ \phi ( x) = \mathop{\rm ln} x $ and $ c $ is the Euler constant. The formula has also been generalized to multiple sums.

The Euler–MacLaurin formula finds application in the approximate calculation of definite integrals, the study of convergence of series, the computation of sums, and the expansion of functions in Taylor series. For example, for $ m = 1 $, $ p = 0 $, $ n = 2m + 1 $, and $ \phi ( x) = \cos ( x t - t / 2 ) $, it yields the expression

$$ \frac{t}{2} \mathop{\rm cotan} \frac{t}{2} = \sum _ {\nu = 0 } ^ { m } (- 1) ^ \nu \frac{t ^ {2 \nu } }{( 2 \nu ) ! } B _ {2 \nu } + $$

$$ + \frac{( - 1 ) ^ {m + 1 } t ^ {2m+ 2} }{2 \ \sin ( t / 2) } \int\limits _ { 0 } ^ { 1 } \frac{B _ {2m+ 1} ( t) }{( 2 m + 1 ) ! } \sin \left ( x - \frac{1}{2} \right ) t dx . $$

The Euler–MacLaurin formula plays an important role in the study of asymptotic expansions, number-theoretic estimates and finite-difference calculus.

Sometimes the Euler–MacLaurin formula is applied in the form

$$ \sum _ { 0 } ^ { n } \phi _ {n} ( x) = \int\limits _ { 0 } ^ { n } \phi ( x) \ dx + \frac{1}{2} ( \phi _ {0} + \phi _ {n} ) + $$

$$ + \int\limits _ { 0 } ^ { n } \left ( x - [ x] - \frac{1}{2} \right ) \phi ^ \prime ( x) dx . $$

The formula was first obtained by L. Euler [1] as

$$ S = \int\limits t dn + \alpha t + \beta \frac{dt}{dn} + \gamma \frac{d ^ {2} t }{d n ^ {2} } + \delta \frac{d ^ {2} t }{d n ^ {2} } + \epsilon \frac{d ^ {4} t }{d n ^ {4} } + \dots , $$

where $ S $ is the sum of the first terms of the series with general term $ t ( n) $, $ S = t = 0 $ for $ n = 0 $, and the coefficients are determined from the recurrence relations

$$ \alpha = \frac{1}{2} ,\ \beta = \frac \alpha {2!} - \frac{1}{3!} = \frac{1}{12} ,\ \gamma = \frac \beta {2!} - \frac \alpha {3!} + \frac{1}{4!} = 0 , $$

$$ \delta = \frac \gamma {2!} - \frac \beta {3!} + \frac \alpha {4!} - \frac{1}{5!} = - \frac{1}{720} ,\ \epsilon = 0 ,\ \gamma = 0 ,\dots . $$

The formula was later discovered independently by C. MacLaurin [2].

References

[1] L. Euler, Comment. Acad. Sci. Imp. Petrop. , 6 (1738) pp. 68–97
[2] C. MacLaurin, "A treatise of fluxions" , 1–2 , Edinburgh (1742)
[3] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[4] N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924)
[5] A.O. [A.O. Gel'fond] Gelfond, "Differenzenrechnung" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)

Comments

The use of the Euler–MacLaurin sum formula in numerical quadrature is discussed in [a1] and [a2]. By replacing the various derivatives by finite differences the quadrature rules of Bessel, Gauss and Gregory are obtained.

References

[a1] F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)
[a2] J.F. Steffensen, "Interpolation" , Chelsea, reprint (1950)
How to Cite This Entry:
Euler-MacLaurin formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler-MacLaurin_formula&oldid=22393
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article