Euler-MacLaurin formula

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)