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1: 4.11 Sums
For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).
2: 4.22 Infinite Products and Partial Fractions
§4.22 Infinite Products and Partial Fractions
3: 4.36 Infinite Products and Partial Fractions
§4.36 Infinite Products and Partial Fractions
4: 25.8 Sums
25.8.1 k = 2 ( ζ ( k ) 1 ) = 1 .
25.8.2 k = 0 Γ ( s + k ) ( k + 1 ) ! ( ζ ( s + k ) 1 ) = Γ ( s 1 ) , s 1 , 0 , 1 , 2 , .
25.8.5 k = 2 ζ ( k ) z k = γ z z ψ ( 1 z ) , | z | < 1 .
25.8.9 k = 1 ζ ( 2 k ) ( 2 k + 1 ) 2 2 k = 1 2 1 2 ln 2 .
25.8.10 k = 1 ζ ( 2 k ) ( 2 k + 1 ) ( 2 k + 2 ) 2 2 k = 1 4 7 4 π 2 ζ ( 3 ) .
5: 25.2 Definition and Expansions
25.2.1 ζ ( s ) = n = 1 1 n s .
§25.2(ii) Other Infinite Series
25.2.6 ζ ( s ) = n = 2 ( ln n ) n s , s > 1 .
§25.2(iv) Infinite Products
25.2.11 ζ ( s ) = p ( 1 p s ) 1 , s > 1 ,
6: Sidebar 5.SB1: Gamma & Digamma Phase Plots
This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. …
7: 5.8 Infinite Products
§5.8 Infinite Products
8: 1.9 Calculus of a Complex Variable
§1.9(v) Infinite Sequences and Series
This sequence converges pointwise to a function f ( z ) if …
§1.9(vii) Inversion of Limits
Dominated Convergence Theorem
9: 24.8 Series Expansions
§24.8(i) Fourier Series
If n = 1 , 2 , and 0 x 1 , then …
§24.8(ii) Other Series
24.8.9 E 2 n = ( 1 ) n k = 1 k 2 n cosh ( 1 2 π k ) 4 k = 0 ( 1 ) k ( 2 k + 1 ) 2 n e 2 π ( 2 k + 1 ) 1 , n = 1 , 2 , .
10: 23.17 Elementary Properties
§23.17(iii) Infinite Products