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1: 1.5 Calculus of Two or More Variables
and the second-order term in (1.5.18) is positive definite (negative definite), that is, …
2: 25.11 Hurwitz Zeta Function
25.11.30 ζ ( s , a ) = Γ ( 1 - s ) 2 π i - ( 0 + ) e a z z s - 1 1 - e z d z , s 1 , a > 0 ,
where the integration contour is a loop around the negative real axis as described for (25.5.20). …
25.11.32 0 a x n ψ ( x ) d x = ( - 1 ) n - 1 ζ ( - n ) + ( - 1 ) n H n B n + 1 n + 1 - k = 0 n ( - 1 ) k ( n k ) H k B k + 1 ( a ) k + 1 a n - k + k = 0 n ( - 1 ) k ( n k ) ζ ( - k , a ) a n - k , n = 1 , 2 , , a > 0 ,
25.11.34 n 0 a ζ ( 1 - n , x ) d x = ζ ( - n , a ) - ζ ( - n ) + B n + 1 - B n + 1 ( a ) n ( n + 1 ) , n = 1 , 2 , , a > 0 .
3: 33.14 Definitions and Basic Properties
When ϵ < 0 and > ( - ϵ ) - 1 / 2 the quantity A ( ϵ , ) may be negative, causing s ( ϵ , ; r ) and c ( ϵ , ; r ) to become imaginary. …
33.14.15 0 ϕ m , ( r ) ϕ n , ( r ) d r = δ m , n .