About the Project
NIST

Euler–Maclaurin formula

AdvancedHelp

(0.001 seconds)

1—10 of 14 matching pages

1: 24.17 Mathematical Applications
EulerMaclaurin Summation Formula
2: 25.2 Definition and Expansions
§25.2(iii) Representations by the EulerMaclaurin Formula
25.2.8 ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - s N x - x x s + 1 d x , s > 0 , N = 1 , 2 , 3 , .
25.2.9 ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - 1 2 N - s + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k N 1 - s - 2 k - ( s + 2 n 2 n + 1 ) N B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n ; n , N = 1 , 2 , 3 , .
25.2.10 ζ ( s ) = 1 s - 1 + 1 2 + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k - ( s + 2 n 2 n + 1 ) 1 B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n , n = 1 , 2 , 3 , .
3: 2.10 Sums and Sequences
§2.10(i) EulerMaclaurin Formula
This is the EulerMaclaurin formula. … In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at s = 2 m - 1 , where m is any positive integer satisfying m 1 2 ( α + 1 ) . For extensions of the EulerMaclaurin formula to functions f ( x ) with singularities at x = a or x = n (or both) see Sidi (2004). …
4: Bibliography E
  • D. Elliott (1998) The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).
  • 5: 25.11 Hurwitz Zeta Function
    §25.11(iii) Representations by the EulerMaclaurin Formula
    25.11.5 ζ ( s , a ) = n = 0 N 1 ( n + a ) s + ( N + a ) 1 - s s - 1 - s N x - x ( x + a ) s + 1 d x , s 1 , s > 0 , a > 0 , N = 0 , 1 , 2 , 3 , .
    25.11.6 ζ ( s , a ) = 1 a s ( 1 2 + a s - 1 ) - s ( s + 1 ) 2 0 B ~ 2 ( x ) - B 2 ( x + a ) s + 2 d x , s 1 , s > - 1 , a > 0 .
    25.11.7 ζ ( s , a ) = 1 a s + 1 ( 1 + a ) s ( 1 2 + 1 + a s - 1 ) + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k 1 ( 1 + a ) s + 2 k - 1 - ( s + 2 n 2 n + 1 ) 1 B ~ 2 n + 1 ( x ) ( x + a ) s + 2 n + 1 d x , s 1 , a > 0 , n = 1 , 2 , 3 , , s > - 2 n .
    6: Bibliography H
  • M. Hauss (1997) An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to ζ ( 2 m + 1 ) . Commun. Appl. Anal. 1 (1), pp. 15–32.
  • 7: 3.5 Quadrature
    If k in (3.5.4) is not arbitrarily large, and if odd-order derivatives of f are known at the end points a and b , then the composite trapezoidal rule can be improved by means of the EulerMaclaurin formula2.10(i)). …
    8: Bibliography B
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • 9: 9.12 Scorer Functions
    §9.12(v) Connection Formulas
    §9.12(vi) Maclaurin Series
    where the integration contour separates the poles of Γ ( 1 3 + 1 3 t ) from those of Γ ( - t ) . … For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … where γ is Euler’s constant (§5.2(ii)). …
    10: 13.2 Definitions and Basic Properties
    The first two standard solutions are: … Although M ( a , b , z ) does not exist when b = - n , n = 0 , 1 , 2 , , many formulas containing M ( a , b , z ) continue to apply in their limiting form. …
    13.2.18 U ( a , b , z ) = Γ ( b - 1 ) Γ ( a ) z 1 - b + Γ ( 1 - b ) Γ ( a - b + 1 ) + O ( z 2 - b ) , 1 b < 2 , b 1 ,
    13.2.19 U ( a , 1 , z ) = - 1 Γ ( a ) ( ln z + ψ ( a ) + 2 γ ) + O ( z ln z ) ,
    §13.2(vii) Connection Formulas