limit points (or limiting points)

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11—20 of 41 matching pages

11: 19.14 Reduction of General Elliptic Integrals

… ►A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …

12: 19.29 Reduction of General Elliptic Integrals

… ►Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. …

13: 2.10 Sums and Sequences

… ►To match the limiting behavior of

f(z)

at these points we set …

14: 1.5 Calculus of Two or More Variables

… ►

§1.5(i) Partial Derivatives

… ►A function is continuous on a point set

D

if it is continuous at all points of

D

. … … ►Sufficient conditions for the limit to exist are that

f(x,y)

is continuous, or piecewise continuous, on

R

. … ►Again the mapping is one-to-one except perhaps for a set of points of volume zero. …

15: 1.4 Calculus of One Variable

… ►

1.4.18

\int^{b}_{a}f(x)\,\mathrm{d}x=\lim\sum^{n-1}_{j=0}f(\xi_{j})(x_{j+1}-x_{j})

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $j$ : integer, $n$ : nonnegative integer and $\xi_{\NVar{j}}$ : point
Referenced by:: §1.4(v)
Permalink:: http://dlmf.nist.gov/1.4.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

…

16: 15.6 Integral Representations

… ►In (15.6.3) the point

\ifrac{1}{(z-1)}

lies outside the integration contour, the contour cuts the real axis between

t=-1

and

0

, at which point

\operatorname{ph}t=\pi

and

\operatorname{ph}\left(1+t\right)=0

. ►In (15.6.4) the point

\ifrac{1}{z}

lies outside the integration contour, and at the point where the contour cuts the negative real axis

\operatorname{ph}t=\pi

and

\operatorname{ph}\left(1-t\right)=0

. ►In (15.6.5) the integration contour starts and terminates at a point

A

on the real axis between

0

and

1

. …At the starting point

\operatorname{ph}t

and

\operatorname{ph}\left(1-t\right)

are zero. If desired, and as in Figure 5.12.3, the upper integration limit in (15.6.5) can be replaced by

(1+,0+,1-,0-)

. …

17: 15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►Here

\{a_{1},a_{2}\}

\{b_{1},b_{2}\}

\{c_{1},c_{2}\}

are the exponent pairs at the points

\alpha

\beta

\gamma

, respectively. …Also, if any of

\alpha

\beta

\gamma

, is at infinity, then we take the corresponding limit in (15.11.1). … ►

15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►These constants can be chosen to map any two sets of three distinct points

\{\alpha,\beta,\gamma\}

and

\{\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\}

onto each other. …

18: 14.24 Analytic Continuation

… ►the limiting value being taken in (14.24.1) when

2\nu

is an odd integer. ►Next, let

P^{-\mu}_{\nu,s}\left(z\right)

and

\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)

denote the branches obtained from the principal branches by encircling the branch point

1

(but not the branch point

-1

)

s

times in the positive sense. …the limiting value being taken in (14.24.4) when

\mu\in\mathbb{Z}

. …

19: 10.72 Mathematical Applications

… ►

§10.72(i) Differential Equations with Turning Points

… ►

Simple Turning Points

… ►These expansions are uniform with respect to

z

, including the turning point

z_{0}

and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … ►

Multiple or Fractional Turning Points

… ►

§10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point

…

20: 18.40 Methods of Computation

… ►It is now necessary to take the limit

\varepsilon\to 0{+}

F(x^{\prime}+\mathrm{i}\varepsilon)

, and the imaginary part is the required Stieltjes–Perron inversion: …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. … ►The quadrature points and weights can be put to a more direct and efficient use. … ►This allows Stieltjes–Perron inversion for the

w(x_{i,N})

, given the quadrature weights and points. … ►

See accompanying text — Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. …For the derivative rule Lagrange interpolation (red points) gives $\sim 15$ digits in the central region, while PWCF interpolation (blue points) gives $\sim 25$ . Magnify

…