punctured neighborhood

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1: 1.10 Functions of a Complex Variable

… ►Let $r_1=0$, so that the annulus becomes the punctured neighborhood $N$: , and assume that $f(z)$ is analytic in $N$, but not at $z_0$. …Lastly, if $a_n\ne 0$ for infinitely many negative $n$, then $z_0$ is an isolated essential singularity. … ►A cut neighborhood is formed by deleting a ray emanating from the center. … ►Suppose $F(z)$ is multivalued and $a$ is a point such that there exists a branch of $F(z)$ in a cut neighborhood of $a$, but there does not exist a branch of $F(z)$ in any punctured neighborhood of $a$. …

2: 2.7 Differential Equations

… ►In a punctured neighborhood $𝐍$ of a regular singularity $z_0$ … ►

2.7.4 $$w_j(z)={(z-z_0)}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}{a}_{s,j}{(z-z_0)}^s,$$ $z\in 𝐍$,

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Symbols:

$\in$: element of, $𝐍$: punctured neighborhood, $w_j(z)$: solutions and $a_n$: coefficients

Referenced by:

§2.7(i), §2.7(i)

Permalink:

http://dlmf.nist.gov/2.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(i), §2.7 and Ch.2

… ►

2.7.6 $$w_2(z)={(z-z_0)}^{{\alpha }_2}\sum _{\begin{array}{c}s=0\\ s\ne {\alpha }_1-{\alpha }_2\end{array}}^{\mathrm{\infty }}{b}_s{(z-z_0)}^s+c{w}_1(z)\ln \left(z-z_0\right),$$ $z\in 𝐍$.

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Symbols:

$\in$: element of, $\ln z$: principal branch of logarithm function, $b_s$: coefficients, $c$: constant, $𝐍$: punctured neighborhood and $w_j(z)$: solutions

Referenced by:

§2.7(i), §2.9(ii)

Permalink:

http://dlmf.nist.gov/2.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(i), §2.7 and Ch.2

… ►In a neighborhood, or sectorial neighborhood of a singularity, one member has to be recessive. …

3: 9.16 Physical Applications

… ► Airy on the intensity of light in the neighborhood of a caustic (Airy (1838, 1849)). … ►The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. … ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. … ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …

4: 31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

5: 31.18 Methods of Computation

… ►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …

6: 10.72 Mathematical Applications

… ►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. … ►These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of $z_0$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … ►These approximations are uniform with respect to both $z$ and $\alpha$, including $z=z_0(a)$, the cut neighborhood of $z=0$, and $\alpha =a$. …

7: 1.9 Calculus of a Complex Variable

… ►

Point Sets in $ℂ$

… ►An open set in $ℂ$ is one in which each point has a neighborhood that is contained in the set. … ►A system of open disks around infinity is given by …Each $S_r$ is a neighborhood of $\mathrm{\infty }$. Also, …

8: 3.8 Nonlinear Equations

… ►An iterative method converges locally to a solution $\zeta$ if there exists a neighborhood $N$ of $\zeta$ such that $z_n\to \zeta$ whenever the initial approximation $z_0$ lies within $N$. … ►Starting this iteration in the neighborhood of one of the four zeros $\pm 1,\pm \mathrm{i}$, sequences $\{z_n\}$ are generated that converge to these zeros. …

9: 10.2 Definitions

… ►

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

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Pair	Interval or Region
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$J_{\nu }\left(z\right),Y_{\nu }\left(z\right)$	neighborhood of 0 in $|\mathrm{ph}z|\le \pi$
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$H_{\nu }^{(1)}\left(z\right),H_{\nu }^{(2)}\left(z\right)$	neighborhood of $\mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi$

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10: 2.3 Integrals of a Real Variable

… ►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … ►derives from the neighborhood of the minimum of $p(t)$ in the integration range. … ►

$p^{\prime }(t)$ and $q(t)$ are continuous in a neighborhood of $a$, save possibly at $a$, and the minimum of $p(t)$ in $[a,b)$ is approached only at $a$.

… ►However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of $p^{\prime }(t)$ because $p(t)$ changes relatively slowly at these stationary points. …