nonuniformity of convergence

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31: 35.2 Laplace Transform

… ►Then (35.2.1) converges absolutely on the region

\Re\left(\mathbf{Z}\right)>\mathbf{X}_{0}

, and

g(\mathbf{Z})

is a complex analytic function of all elements

z_{j,k}

\mathbf{Z}

. … ►Assume that

\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{U}+\mathrm{i}\mathbf{V})\right|% \,\mathrm{d}{\mathbf{V}}

converges, and also that its limit as

\mathbf{U}\to\infty

0

. …

32: 18.40 Methods of Computation

… ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to

w(x)

, as will be considered in the following paragraphs. … ►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

\Im{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. …Gautschi (2004, p. 119–120) has explored the

\varepsilon\to 0^{+}

limit via the Wynn

\varepsilon

-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in

w(x)

, depending smoothly on

x^{\prime}

, for

N\approx 4000

, for an example involving first numerator Legendre OP’s. … ►Convergence is

\sim O\left(N^{-2}\right)

. … ►Further, exponential convergence in

N

, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate

w(x)

for these OP systems on

x\in[-1,1]

and

(-\infty,\infty)

respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

33: 31.11 Expansions in Series of Hypergeometric Functions

… ►and (31.11.1) converges to (31.3.10) outside the ellipse

\mathcal{E}

in the

z

-plane with foci at 0, 1, and passing through the third finite singularity at

z=a

. ►Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function. … ►The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities

z=0

and

z=1

. … ►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse

\mathcal{E}

. … ►For Heun functions (§31.4) they are convergent inside the ellipse

\mathcal{E}

. …

34: 1.4 Calculus of One Variable

… ►When the limits in (1.4.22) and (1.4.23) exist, the integrals are said to be convergent. …Absolute convergence also implies convergence. … ► … ► …

35: 28.6 Expansions for Small $q$

… ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. … ►

Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.

► ►►

$n$	$\rho^{(1)}_{n}$		$\rho^{(2)}_{n}$		$\rho^{(3)}_{n}$
…

► … ►

28.6.20

\liminf_{n\to\infty}\frac{\rho_{n}^{(j)}}{n^{2}}\geq kk^{\prime}(K\left(k% \right))^{2}=2.04183\;4\dots,

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\liminf$ : least limit point, $n$ : integer, $j$ : integer, $\rho_{n}^{(j)}$ : radii of convergence and $k$ : root
Referenced by:: §28.6(i), §28.6(ii)
Permalink:: http://dlmf.nist.gov/28.6.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

… ►The radii of convergence of the series (28.6.21)–(28.6.26) are the same as the radii of the corresponding series for

a_{n}\left(q\right)

and

b_{n}\left(q\right)

; compare Table 28.6.1 and (28.6.20).

36: 15.2 Definitions and Analytical Properties

… ►On the circle of convergence,

|z|=1

, the Gauss series: ►

(a)

Converges absolutely when $\Re\left(c-a-b\right)>0$ .

►

(b)

Converges conditionally when $-1<\Re\left(c-a-b\right)\leq 0$ and $z=1$ is excluded.

…

37: 16.2 Definition and Analytic Properties

… ►When

p\leq q

the series (16.2.1) converges for all finite values of

z

and defines an entire function. … ►If none of the

a_{j}

is a nonpositive integer, then the radius of convergence of the series (16.2.1) is

1

, and outside the open disk

|z|<1

the generalized hypergeometric function is defined by analytic continuation with respect to

z

. … ►On the circle

|z|=1

the series (16.2.1) is absolutely convergent if

\Re\gamma_{q}>0

, convergent except at

z=1

-1<\Re\gamma_{q}\leq 0

, and divergent if

\Re\gamma_{q}\leq-1

, where …

38: 1.16 Distributions

… ► ►The linear space of all test functions with the above definition of convergence is called a test function space. … ►We say that a sequence of distributions

\{\Lambda_{n}\}

converges to a distribution

\Lambda

\mathcal{D}^{*}

if … ►A sequence

\{\phi_{n}\}

of functions in

\mathcal{T}

is said to converge to a function

\phi\in\mathcal{T}

n\to\infty

if the sequence

\{\phi_{n}^{(k)}\}

converges uniformly to

\phi^{(k)}

on every finite interval and if the constants

c_{k,N}

in the inequalities … …

39: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

40: 16.17 Definition

… ►

(i)

$L$ goes from $-\mathrm{i}\infty$ to $\mathrm{i}\infty$ . The integral converges if $p+q<2(m+n)$ and $|\operatorname{ph}z|<(m+n-\frac{1}{2}(p+q))\pi$ .

►

(ii)

$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\Gamma\left(b_{\ell}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ( $\neq 0$ ) if $p<q$ , and for $0<|z|<1$ if $p=q\geq 1$ .

►

(iii)

$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\Gamma\left(1-a_{\ell}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all $z$ if $p>q$ , and for $|z|>1$ if $p=q\geq 1$ .

…