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1: 18.40 Methods of Computation
The question is then: how is this possible given only F N ( z ) , rather than F ( z ) itself? F N ( z ) often converges to smooth results for z off the real axis for z at a distance greater than the pole spacing of the x n , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to F N ( z ) and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …
2: 1.9 Calculus of a Complex Variable
This sequence converges pointwise to a function f ( z ) if …
3: 18.2 General Orthogonal Polynomials
This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with p n ( x ) = T n ( x ) , a Chebyshev polynomial of the first kind, see Table 18.3.1. …
4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where f ( x ) is continuous, with convergence to ( f ( x 0 ) + f ( x 0 + ) ) / 2 if x 0 is an isolated point of discontinuity. …