By Arnold R. Krommer
This article discusses computational integration equipment and the elemental mathematical ideas they're in response to. It comprises sections on one-dimensional and multi-dimensional integration formulation, and it bargains with matters in regards to the building of numerical integration algorithms.
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160) = 1 2π 1 2π π −π π √ cos(ν(κ)t)eiκn dκ = J2|n| ( 2t), sin(ν(κ)t) iκn e dκ = ν(κ) −π 2|n|+1 t cn (s)ds 0 2|n| + 1 2|n| + 3 t t2 F ; ( , 2|n| + 1); − ( ). 1 2 2 2 2 2|n| (|n| + 1)! Here Jn (x), p Fq (u; v; x) denote the Bessel and generalized hypergeometric functions, respectively. From this form one can deduce that a localized wave (say compactly supported at t = 0) will spread as t increases (cf. 14). This phenomenon is due to the fact that different plane waves travel with different speed and is called dispersion.
P. at +∞. c. 164) n∈N 1 2 ≤ |a(n)| |a(0)| |c(z, n)|2 n∈N |s(z, n)|2 .
4. 124) τˆf (n) = f (n + 1) + f (n − 1) + d(n)f (n) , w(n) where w(n) > 0, d(n) ∈ R, and (w(n)w(n + 1))−1 , w(n)−1 d(n) are bounded ˆ called Helmholtz operator, in the sequences. 125) f, g = 2 f, g ∈ w(n)f (n)g(n), (Z; w). 20) holds with little modifications and H bounded and self-adjoint. There is an interesting connection between Jacobi and Helmholtz operators stated in the next theorem. 14. Let H be the Jacobi operator associated with the sequences a(n) > ˆ be the Helmholtz operator associated with the sequences w(n) > 0, 0, b(n) and let H d(n).