convergence properties

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11: 28.31 Equations of Whittaker–Hill and Ince

… ►For proofs and further information, including convergence of the series (28.31.4), (28.31.5), see Arscott (1967). … ►

$${\mathrm{ℎ𝑠}}_{2n+2}^{2m+2}(z,-\xi )={(-1)}^m{\mathrm{ℎ𝑠}}_{2n+2}^{2m+2}(\frac{1}{2}\pi -z,\xi ).$$

… ►More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by …

12: 16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. … ►Then the integral converges when provided that $z\ne 0$, or when $p=q+1$ provided that , and provides an integral representation of the left-hand side with these conditions. … ►Then the integral converges when and . …

13: 13.2 Definitions and Basic Properties

§13.2 Definitions and Basic Properties

… ►The series (13.2.2) and (13.2.3) converge for all $z\in ℂ$. … … ►Another standard solution of (13.2.1) is $U(a,b,z)$, which is determined uniquely by the property …

14: 20.11 Generalizations and Analogs

… ►For applications to rapidly convergent expansions for $\pi$ see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …

15: 18.38 Mathematical Applications

… ►The monic Chebyshev polynomial $2^{1-n}{T}_n\left(x\right)$, $n\ge 1$, enjoys the ‘minimax’ property on the interval $[-1,1]$, that is, $|2^{1-n}{T}_n\left(x\right)|$ has the least maximum value among all monic polynomials of degree $n$. In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. …

16: 33.14 Definitions and Basic Properties

§33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

… ►This includes $ϵ=0$, hence $f(ϵ,\mathrm{\ell };r)$ can be expanded in a convergent power series in $ϵ$ in a neighborhood of $ϵ=0$ (§33.20(ii)). ►

§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

… ►The function $s(ϵ,\mathrm{\ell };r)$ has the following properties: …

17: 8.17 Incomplete Beta Functions

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§8.17(i) Definitions and Basic Properties

… ►The $4m$ and $4m+1$ convergents are less than $I_x(a,b)$, and the $4m+2$ and $4m+3$ convergents are greater than $I_x(a,b)$. … ►The expansion (8.17.22) converges rapidly for . For $x>(a+1)/(a+b+2)$ or , more rapid convergence is obtained by computing $I_{1-x}(b,a)$ and using (8.17.4). … ►

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

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18: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large $x$ or $|z|$, whether or not $\nu$ is large. … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

19: 3.11 Approximation Techniques

… ►The iterative process converges locally and quadratically (§3.8(i)). … ►They enjoy an orthogonal property with respect to integrals: … ►converges uniformly. … ► … ►The property …

20: 3.6 Linear Difference Equations

… ►The normalizing factor $\mathrm{\Lambda }$ can be the true value of $w_0$ divided by its trial value, or $\mathrm{\Lambda }$ can be chosen to satisfy a known property of the wanted solution of the form … ►For further information on Miller’s algorithm, including examples, convergence proofs, and error analyses, see Wimp (1984, Chapter 4), Gautschi (1967, 1997b), and Olver (1964a). … ►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …