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1: 1.17 Integral and Series Representations of the Dirac Delta
§1.17(i) Delta Sequences
for a suitably chosen sequence of functions δ n ( x ) , n = 1 , 2 , . Such a sequence is called a delta sequence and we write, symbolically,
1.17.4 lim n δ n ( x ) = δ ( x ) , x .
An example of a delta sequence is provided by …
2: 3.9 Acceleration of Convergence
§3.9(iii) Aitken’s Δ 2 -Process
3.9.7 t n = s n - ( Δ s n ) 2 Δ 2 s n = s n - ( s n + 1 - s n ) 2 s n + 2 - 2 s n + 1 + s n .
3.9.9 t n , 2 k = H k + 1 ( s n ) H k ( Δ 2 s n ) , n = 0 , 1 , 2 , ,
3: 30.15 Signal Analysis
30.15.7 - τ τ ϕ k ( t ) ϕ n ( t ) d t = Λ n δ k , n ,
30.15.8 - ϕ k ( t ) ϕ n ( t ) d t = δ k , n .
The sequence ϕ n , n = 0 , 1 , 2 , forms an orthonormal basis in the space of σ -bandlimited functions, and, after normalization, an orthonormal basis in L 2 ( - τ , τ ) . …
4: 3.6 Linear Difference Equations
3.6.2 a n Δ 2 w n - 1 + ( 2 a n - b n ) Δ w n - 1 + ( a n - b n + c n ) w n - 1 = d n ,
5: 3.8 Nonlinear Equations
Let z 1 , z 2 , be a sequence of approximations to a root, or fixed point, ζ . …for all n sufficiently large, where A and p are independent of n , then the sequence is said to have convergence of the p th order. … For real functions f ( x ) the sequence of approximations to a real zero ξ will always converge (and converge quadratically) if either: … Starting this iteration in the neighborhood of one of the four zeros ± 1 , ± i , sequences { z n } are generated that converge to these zeros. For an arbitrary starting point z 0 , convergence cannot be predicted, and the boundary of the set of points z 0 that generate a sequence converging to a particular zero has a very complicated structure. …
6: 17.12 Bailey Pairs
17.12.1 n = 0 α n γ n = n = 0 β n δ n ,
γ n = j = n δ j u j - n v j + n .
A sequence of pairs of rational functions of several variables ( α n , β n ) , n = 0 , 1 , 2 , , is called a Bailey pair provided that for each n 0 When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …
7: 4.4 Special Values and Limits
4.4.16 lim z z a e - z = 0 , | ph z | 1 2 π - δ ( < 1 2 π ),
where a ( ) and δ ( ( 0 , 1 2 π ] ) are constants. …
4.4.19 lim n ( ( k = 1 n 1 k ) - ln n ) = γ = 0.57721 56649 01532 86060 ,
8: 18.27 q -Hahn Class
18.27.2 x X p n ( x ) p m ( x ) | x | v x = h n δ n , m ,
Here a , b are fixed positive real numbers, and I + and I - are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. …
18.27.4 y = 0 N Q n ( q - y ) Q m ( q - y ) ( α q , q - N ; q ) y ( α β q ) - y ( q , β - 1 q - N ; q ) y = h n δ n , m , n , m = 0 , 1 , , N .
18.27.14 y = 0 p n ( q y ) p m ( q y ) ( b q ; q ) y ( a q ) y ( q ; q ) y = h n δ n , m , 0 < a < q - 1 , b < q - 1 .
18.27.16 0 L n ( α ) ( x ; q ) L m ( α ) ( x ; q ) x α ( - x ; q ) d x = ( q α + 1 ; q ) n ( q ; q ) n q n h 0 ( 1 ) δ n , m , α > - 1 ,
9: 5.17 Barnes’ G -Function (Double Gamma Function)
When z in | ph z | π - δ ( < π ) , …
5.17.6 A = e C = 1.28242 71291 00622 63687 ,
10: 1.16 Distributions
A sequence { ϕ n } of test functions converges to a test function ϕ if the support of every ϕ n is contained in a fixed compact set K and as n the sequence { ϕ n ( k ) } converges uniformly on K to ϕ ( k ) for k = 0 , 1 , 2 , . … We say that a sequence of distributions { Λ n } converges to a distribution Λ in 𝒟 * if …
§1.16(iii) Dirac Delta Distribution
The Dirac delta distribution is singular. … A sequence of tempered distributions Λ n converges to Λ in 𝒯 * if …