point sets in complex plane

(0.012 seconds)

11—20 of 44 matching pages

11: 22.4 Periods, Poles, and Zeros

… ►For each Jacobian function, Table 22.4.1 gives its periods in the

z

-plane in the left column, and the position of one of its poles in the second row. The other poles are at congruent points, which is the set of points obtained by making translations by

2mK+2niK^{\prime}

, where

m,n\in\mathbb{Z}

. … ►Table 22.4.2 displays the periods and zeros of the functions in the

z

-plane in a similar manner to Table 22.4.1. … ►The other poles and zeros are at the congruent points. … ►The set of points

z=mK+niK^{\prime}

m,n\in\mathbb{Z}

, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by

mK+niK^{\prime}

, where again

m,n\in\mathbb{Z}

. …

12: 1.10 Functions of a Complex Variable

… ►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … ► … ►(a) By introducing appropriate cuts from the branch points and restricting

F(z)

to be single-valued in the cut plane (or domain). … ►Branches of

F(z)

can be defined, for example, in the cut plane

D

obtained from

\mathbb{C}

by removing the real axis from

1

\infty

and from

-1

-\infty

; see Figure 1.10.1. … ►Alternatively, take

z_{0}

to be any point in

D

and set

F(z_{0})={\mathrm{e}}^{\alpha\ln\left(1-z_{0}\right)}{\mathrm{e}}^{\beta\ln% \left(1+z_{0}\right)}

where the logarithms assume their principal values. …

13: 10.20 Uniform Asymptotic Expansions for Large Order

… ►The function

\zeta=\zeta(z)

given by (10.20.2) and (10.20.3) can be continued analytically to the

z

-plane cut along the negative real axis. Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2. ►The equations of the curved boundaries

D_{1}E_{1}

and

D_{2}E_{2}

in the

\zeta

-plane are given parametrically by … ►The curves

BP_{1}E_{1}

and

BP_{2}E_{2}

in the

z

-plane are the inverse maps of the line segments … ►The eye-shaped closed domain in the uncut

z

-plane that is bounded by

BP_{1}E_{1}

and

BP_{2}E_{2}

is denoted by

\mathbf{K}

; see Figure 10.20.3. …

14: 10.2 Definitions

… ►This solution of (10.2.1) is an analytic function of

z\in\mathbb{C}

, except for a branch point at

z=0

when

\nu

is not an integer. The principal branch of

J_{\nu}\left(z\right)

corresponds to the principal value of

(\tfrac{1}{2}z)^{\nu}

(§4.2(iv)) and is analytic in the

z

-plane cut along the interval

(-\infty,0]

. … ►The principal branch corresponds to the principal branches of

J_{\pm\nu}\left(z\right)

in (10.2.3) and (10.2.4), with a cut in the

z

-plane along the interval

(-\infty,0]

. … ►Each solution has a branch point at

z=0

for all

\nu\in\mathbb{C}

. The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the

z

-plane along the interval

(-\infty,0]

. …

15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In particular, this holds for

F(\lambda)=\left(z-\lambda\right)^{-1}

z\in\mathbb{C}\setminus\boldsymbol{\sigma}_{c}

… ►Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies

\lambda_{\mathrm{res}}-\mathrm{i}\Gamma_{\mathrm{res}}/2

corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to

1/\Gamma_{\mathrm{res}}

. … ►The resolvent set

\rho(T)

consists of all

z\in\mathbb{C}

such that (i)

z-T

is injective, (ii)

\mathcal{R}(z-T)

is dense in

V

, (iii) the resolvent

\left(z-T\right)^{-1}

is bounded. … ►

The point spectrum $\boldsymbol{\sigma}_{p}$ . It consists of all $z\in\mathbb{C}$ for which $z-T$ is not injective, or equivalently, for which $z$ is an eigenvalue of $T$ , i.e., $Tv=zv$ for some $v\in\mathcal{D}(T)\backslash\{0\}$ .

… ►Boundary values and boundary conditions for the end point

b

are defined in a similar way. …

16: 2.1 Definitions and Elementary Properties

… ►Let

\mathbf{X}

be a point set with a limit point

c

. As

x\to c

\mathbf{X}

… ►as

x\to\infty

in an unbounded set

\mathbf{X}

\mathbb{R}

\mathbb{C}

. … ►Suppose

u

is a parameter (or set of parameters) ranging over a point set (or sets)

\mathbf{U}

, and for each nonnegative integer

n

… ►Similarly for finite limit point

c

in place of

\infty

. …

17: 1.13 Differential Equations

… ►A domain in the complex plane is simply-connected if it has no “holes”; more precisely, if its complement in the extended plane

\mathbb{C}\cup\{\infty\}

is connected. … ►A solution becomes unique, for example, when

w

and

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

are prescribed at a point in

D

. … ►Suppose also that at (a fixed)

z_{0}\in D

w

and

\ifrac{\partial w}{\partial z}

are analytic functions of

u

. … ►

Transformation of the Point at Infinity

… ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues,

\lambda

; (ii) the corresponding (real) eigenfunctions,

u(x)

and

w(t)

, have the same number of zeros, also called nodes, for

t\in(0,c)

as for

x\in(a,b)

; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

18: 15.11 Riemann’s Differential Equation

… ►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). The most general form is given by … ►In particular, … ►A conformal mapping of the extended complex plane onto itself has the form …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. …

19: 29.2 Differential Equations

… ►This equation has regular singularities at the points

2pK+(2q+1)\mathrm{i}{K^{\prime}}

, where

p,q\in\mathbb{Z}

, and

K

{K^{\prime}}

are the complete elliptic integrals of the first kind with moduli

k

k^{\prime}(=(1-k^{2})^{1/2})

, respectively; see §19.2(ii). In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). … ►

Figure 29.2.1:

z

-plane: singularities

\times

\times

\times

of Lamé’s equation.

… ►

\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})}=k^{2}.

… ►

29.2.8

\eta=(e_{1}-e_{3})^{-1/2}(z-\mathrm{i}{K^{\prime}}),

ⓘ

Defines:: $\eta$ (locally)
Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $k$ : real parameter and $e_{j}$ : constants
Permalink:: http://dlmf.nist.gov/29.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §29.2(ii), §29.2 and Ch.29

…

20: 4.37 Inverse Hyperbolic Functions

… ►In (4.37.1) the integration path may not pass through either of the points

t=\pm i

, and the function

(1+t^{2})^{1/2}

assumes its principal value when

t

is real. In (4.37.2) the integration path may not pass through either of the points

\pm 1

, and the function

(t^{2}-1)^{1/2}

assumes its principal value when

t\in(1,\infty)

. …

\operatorname{Arcsinh}z

and

\operatorname{Arccsch}z

have branch points at

z=\pm i

; the other four functions have branch points at

z=\pm 1

. … ►The principal values (or principal branches) of the inverse

\sinh

\cosh

, and

\tanh

are obtained by introducing cuts in the

z

-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. … ►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. …