nonuniformity of convergence
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21: 33.8 Continued Fractions
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►The continued fraction (33.8.1) converges for all finite values of , and (33.8.2) converges for all .
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►The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision.
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22: 15.15 Sums
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►Here () is an arbitrary complex constant and the expansion converges when .
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23: 9.17 Methods of Computation
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►Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation.
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24: 11.13 Methods of Computation
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►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation.
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25: 1.8 Fourier Series
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►(1.8.10) continues to apply if either or or both are infinite and/or has finitely many singularities in , provided that the integral converges uniformly (§1.5(iv)) at , and the singularities for all sufficiently large .
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§1.8(ii) Convergence
… ►Then the series (1.8.1) converges to the sum …The convergence is non-uniform, however, at points where ; see §6.16(i). … ►For other tests for convergence see Titchmarsh (1962b, pp. 405–410). …26: 28.19 Expansions in Series of Functions
27: 28.30 Expansions in Series of Eigenfunctions
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►Then every continuous -periodic function whose second derivative is square-integrable over the interval can be expanded in a uniformly and absolutely convergent series
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28: 29.20 Methods of Computation
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►The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as .
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29: 13.31 Approximations
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►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985).
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►In Luke (1977a) the following rational approximation is given, together with its rate of convergence.
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