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11: 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 …
12: 2.1 Definitions and Elementary Properties
In (2.1.5) can be replaced by any fixed ray in the sector | ph x | < 1 2 π , or by the whole of the sector | ph x | 1 2 π δ . (Here and elsewhere in this chapter δ is an arbitrary small positive constant.) … Let ϕ s ( x ) , s = 0 , 1 , 2 , , be a sequence of functions defined in 𝐗 such that for each s …Then { ϕ s ( x ) } is an asymptotic sequence or scale. Suppose also that f ( x ) and f s ( x ) satisfy …
13: 25.11 Hurwitz Zeta Function
25.11.40 G n = 0 ( 1 ) n ( 2 n + 1 ) 2 = 0.91596 55941 772 .
As a in the sector | ph a | π δ ( < π ) , with s ( 1 ) and δ fixed, we have the asymptotic expansion … Similarly, as a in the sector | ph a | 1 2 π δ ( < 1 2 π ) , …
14: 19.30 Lengths of Plane Curves
From (19.29.7), with a δ = 1 and b δ = 0 , …
19.30.13 P = 4 2 a 2 R F ( 0 , 1 , 2 ) = 2 a 2 × 5.24411 51 = 4 a K ( 1 / 2 ) = a × 7.41629 87 .
15: 18.2 General Orthogonal Polynomials
For such a system, functions f L w 2 ( ( a , b ) ) and sequences { λ n } ( n = 0 , 1 , 2 , ) satisfying n = 0 h n | λ n | 2 < can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2): … The Hankel determinant Δ n of order n is defined by Δ 0 = 1 and …Also define determinants Δ n by Δ 0 = 0 , Δ 1 = μ 1 and … The operator D x is a delta operator, i. … …
16: 18.33 Polynomials Orthogonal on the Unit Circle
18.33.17 | z | = 1 Φ n ( z ) Φ m ( z ) ¯ d μ ( z ) = κ n 2 δ n , m ,
This states that for any sequence { α n } n = 0 with α n and | α n | < 1 the polynomials Φ n ( z ) generated by the recurrence relations (18.33.23), (18.33.24) with Φ 0 ( z ) = 1 satisfy the orthogonality relation (18.33.17) for a unique probability measure μ with infinite support on the unit circle. …
17: 1.12 Continued Fractions
b 0 + a 1 b 1 + a 2 b 2 + is equivalent to b 0 + a 1 b 1 + a 2 b 2 + if there is a sequence { d n } n = 0 , d 0 = 1 ,
d n 0 , such that … A sequence { C n } in the extended complex plane, { } , can be a sequence of convergents of the continued fraction (1.12.3) iff …
1.12.26 1 2 π + δ < ph b n < 1 2 π δ , n = 1 , 2 , 3 , ,
where δ is an arbitrary small positive constant. …
1.12.27 1 2 π + δ < ph C n < 1 2 π δ , n = 1 , 2 , 3 , ,