on a point set

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11—20 of 113 matching pages

11: 1.13 Differential Equations

… ►A solution becomes unique, for example, when

w

and

\ifrac{\mathrm{d}w}{\mathrm{d}z}

are prescribed at a point in

D

. …

12: 3.8 Nonlinear Equations

… ►For an arbitrary starting point

z_{0}\in\mathbb{C}

, convergence cannot be predicted, and the boundary of the set of points

z_{0}

that generate a sequence converging to a particular zero has a very complicated structure. …

13: 23.20 Mathematical Applications

… ►

23.20.4

m=\begin{cases}(3x_{1}^{2}+a)/(2y_{1}),&P_{1}=P_{2},\\ (y_{2}-y_{1})/(x_{2}-x_{1}),&P_{1}\neq P_{2}.\end{cases}

ⓘ

Symbols:: $m$ : integer, $x$ : real part of $z$ , $y$ : imaginary part of $z$ , $a$ : constant and $P$ : points
Permalink:: http://dlmf.nist.gov/23.20.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §23.20(ii), §23.20 and Ch.23

… ►Let

T

denote the set of points on

C

that are of finite order (that is, those points

P

for which there exists a positive integer

n

with

nP=o

), and let

I, K

be the sets of points with integer and rational coordinates, respectively. …

14: 31.3 Basic Solutions

… ►Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. …

15: 4.2 Definitions

… ►In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by

z=x+\mathrm{i}0

, the other set corresponding to the “lower side” and denoted by

z=x-\mathrm{i}0

. …

16: 4.13 Lambert $W$ -Function

… ►

W_{0}\left(z\right)

is a single-valued analytic function on

\mathbb{C}\setminus(-\infty,-{\mathrm{e}}^{-1}]

, real-valued when

z>-{\mathrm{e}}^{-1}

, and has a square root branch point at

z=-{\mathrm{e}}^{-1}

. …The other branches

W_{k}\left(z\right)

are single-valued analytic functions on

\mathbb{C}\setminus(-\infty,0]

, have a logarithmic branch point at

z=0

, and, in the case

k=\pm 1

, have a square root branch point at

z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}

respectively. …

17: 15.6 Integral Representations

… ►In (15.6.2) the point

\ifrac{1}{z}

lies outside the integration contour,

t^{b-1}

and

(t-1)^{c-b-1}

assume their principal values where the contour cuts the interval

(1,\infty)

, and

(1-zt)^{a}=1

t=0

. …

18: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in

\mathbf{x}

space, across which

\Psi_{K}(\mathbf{x};k)

\Psi^{(\mathrm{U})}(\mathbf{x};k)

acquires an exponentially-small asymptotic contribution (in

k

), associated with a complex critical point of

\Phi_{K}

\Phi^{(\mathrm{U})}

. …

19: 36.12 Uniform Approximation of Integrals

§36.12 Uniform Approximation of Integrals

►

§36.12(i) General Theory for Cuspoids

… ►with the

K+1

functions

A(\mathbf{y})

and

\mathbf{x}(\mathbf{y})

determined by correspondence of the

K+1

critical points of

f

and

\Phi_{K}

. … ►For

K=1

, with a single parameter

y

, let the two critical points of

f(u;y)

be denoted by

u_{\pm}(y)

, with

u_{+}>u_{-}

for those values of

y

for which these critical points are real. … ►Also,

\Delta^{1/4}/\sqrt{f_{+}^{\prime\prime}}

and

\Delta^{1/4}/\sqrt{-f_{-}^{\prime\prime}}

are chosen to be positive real when

y

is such that both critical points are real, and by analytic continuation otherwise. …

20: 2.10 Sums and Sequences

… ►As a first estimate for large

n

…for any real constant

\alpha

and the set of all positive integers

j

, we derive … ►The singularities of

f(z)

on the unit circle are branch points at

z=e^{\pm i\alpha}

. To match the limiting behavior of

f(z)

at these points we set …and in the supplementary conditions we may set

m=1

. …