finite expansions

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11: 8.11 Asymptotic Approximations and Expansions

… ►This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$. …

12: 11.10 Anger–Weber Functions

… ►These expansions converge absolutely for all finite values of $z$. …

13: 11.2 Definitions

… ►The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of $z$. …

14: 18.38 Mathematical Applications

… ►The terminology DVR arises as an otherwise continuous variable, such as the co-ordinate $x$, is replaced by its values at a finite set of zeros of appropriate OP’s resulting in expansions using functions localized at these points. …

15: 7.6 Series Expansions

§7.6 Series Expansions

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§7.6(i) Power Series

… ►The series in this subsection and in §7.6(ii) converge for all finite values of $|z|$. ►

§7.6(ii) Expansions in Series of Spherical Bessel Functions

…

16: 2.1 Definitions and Elementary Properties

… ►

§2.1(iii) Asymptotic Expansions

… ►If $c$ is a finite limit point of $𝐗$, then … ►

§2.1(iv) Uniform Asymptotic Expansions

… ►Similarly for finite limit point $c$ in place of $\mathrm{\infty }$. … ►where $c$ is a finite, or infinite, limit point of $𝐗$. …

17: 2.3 Integrals of a Real Variable

… ►Then … ►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. … ►Since $q(t)$ need not be continuous (as long as the integral converges), the case of a finite integration range is included. … ►Then … ►If $p(b)$ is finite, then both endpoints contribute: …

18: 29.19 Physical Applications

… ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …Macfadyen and Winternitz (1971) finds expansions for the two-body relativistic scattering amplitudes. …

19: 18.34 Bessel Polynomials

… ►In this limit the finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty })$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty })$ (see (18.34.5_5)). …

20: 15.15 Sums

… ►Here $z_0$ ($\ne 0$) is an arbitrary complex constant and the expansion converges when $|z-z_0|>\max (|z_0|,|z_0-1|)$. … ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …