points in complex plane

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21: 36.7 Zeros

… ►The zeros in Table 36.7.1 are points in the $𝐱=(x,y)$ plane, where $\mathrm{ph}{\mathrm{\Psi }}_2\left(𝐱\right)$ is undetermined. … ►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the $z$-axis that is far from the origin, the zero contours form an array of rings close to the planes …The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_n$ by at most $(\Delta z)/36$). …In the symmetry planes (e. …There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by …

22: 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 $2mK+2n\mathrm{i}{K}^{\prime }$, 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. … ►Figure 22.4.1 illustrates the locations in the $z$-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices $0$, $2K$, $2K+2\mathrm{i}{K}^{\prime }$, $2\mathrm{i}{K}^{\prime }$. The other poles and zeros are at the congruent points. …

23: 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 $B{P}_1{E}_1$ and $B{P}_2{E}_2$ 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 $B{P}_1{E}_1$ and $B{P}_2{E}_2$ is denoted by $𝐊$; see Figure 10.20.3. …

24: 14.21 Definitions and Basic Properties

… ► $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). The principal branches of $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are real when $\nu$, $\mu \in ℝ$ and $z\in (1,\mathrm{\infty })$. … ►When $\mathrm{\Re }\nu \ge -\frac{1}{2}$ and $\mathrm{\Re }\mu \ge 0$, a numerically satisfactory pair of solutions of (14.21.1) in the half-plane $|\mathrm{ph}z|\le \frac{1}{2}\pi$ is given by $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$. … ►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. …

25: 5.12 Beta Function

… ►In this section all fractional powers have their principal values, except where noted otherwise. … ►with the contour as shown in Figure 5.12.1. … ►When $a,b\in ℂ$ …where the contour starts from an arbitrary point $P$ in the interval $(0,1)$, circles $1$ and then $0$ in the positive sense, circles $1$ and then $0$ in the negative sense, and returns to $P$. It can always be deformed into the contour shown in Figure 5.12.3. …

26: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►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 ${\lambda }_{\mathrm{res}}-\mathrm{i}{\mathrm{\Gamma }}_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. … ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … ►

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

… ►Boundary values and boundary conditions for the end point $b$ are defined in a similar way. … ►See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of $51$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.

27: 2.1 Definitions and Elementary Properties

… ►Let $𝐗$ be a point set with a limit point $c$. As $x\to c$ in $𝐗$ … ►as $x\to \mathrm{\infty }$ in an unbounded set $𝐗$ in $ℝ$ or $ℂ$. … ►For (2.1.14) $𝐗$ can be the positive real axis or any unbounded sector in $ℂ$ of finite angle. … ►Similarly for finite limit point $c$ in place of $\mathrm{\infty }$. …

28: 13.2 Definitions and Basic Properties

… ►In other cases … ►The series (13.2.2) and (13.2.3) converge for all $z\in ℂ$. … ►In particular, … ►In general, $U(a,b,z)$ has a branch point at $z=0$. The principal branch corresponds to the principal value of $z^{-a}$ in (13.2.6), and has a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$; compare §4.2(i). …

29: 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 $ℂ\cup \{\mathrm{\infty }\}$ is connected. … ►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ►

Transformation of the Point at Infinity

… ►in (1.13.1) gives … ►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. …

30: 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.

… ►

29.2.3 $$\xi ={\mathrm{sn}}^2(z,k).$$

ⓘ

Defines:

$\xi$: variable (locally)

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

… ►

29.2.5 $$\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k).$$

ⓘ

Defines:

$\varphi$: change of variable (locally)

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: real parameter

Referenced by:

§29.15(i), §29.15(ii), §29.6(i)

Permalink:

http://dlmf.nist.gov/29.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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