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1: 4.33 Maclaurin Series and Laurent Series
§4.33 Maclaurin Series and Laurent Series
2: 22.10 Maclaurin Series
§22.10 Maclaurin Series
§22.10(i) Maclaurin Series in z
§22.10(ii) Maclaurin Series in k and k
3: 9.4 Maclaurin Series
§9.4 Maclaurin Series
4: 9.17 Methods of Computation
§9.17(i) Maclaurin Expansions
Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of z , they are cumbersome to use when | z | is large owing to slowness of convergence and cancellation. …
5: 5.7 Series Expansions
§5.7(i) Maclaurin and Taylor Series
For 20D numerical values of the coefficients of the Maclaurin series for Γ ( z + 3 ) see Luke (1969b, p. 299). …
6: 4.19 Maclaurin Series and Laurent Series
§4.19 Maclaurin Series and Laurent Series
7: 24.17 Mathematical Applications
Euler–Maclaurin Summation Formula
8: 25.2 Definition and Expansions
§25.2(iii) Representations by the Euler–Maclaurin 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 , .
9: 15.19 Methods of Computation
§15.19(i) Maclaurin Expansions
10: 2.10 Sums and Sequences
§2.10(i) Euler–Maclaurin Formula
This is the Euler–Maclaurin formula. … For extensions of the Euler–Maclaurin formula to functions f ( x ) with singularities at x = a or x = n (or both) see Sidi (2004, 2012b, 2012a). … The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. …