accumulation

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4 matching pages

1: 1.9 Calculus of a Complex Variable

… ►A point

z_{0}

is a limit point (limiting point or accumulation point) of a set of points

S

\mathbb{C}

(or

\mathbb{C}\cup\infty

) if every neighborhood of

z_{0}

contains a point of

S

distinct from

z_{0}

. …Also, the union of

S

and its limit points is the closure of

S

. …

2: 18.39 Applications in the Physical Sciences

… ►

See accompanying text — Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in[-1,1]$ , (18.39.50) for $s=10$ , $l=0$ , and $Z=\pm 1$ . …For $Z=-1$ the weight function, blue curve, is non-zero at $x=-1$ , but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as $k\to\infty$ , $x_{k}\to-1{-}$ . Magnify

… ►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as

x\to-1{-}

. As the Coulomb–Pollaczek OP’s are members of the Nevai-Blumenthal class, this is, for

Z<0

, a physical example of the properties of the zeros of such OP’s, and their possible accumulation at

x=-1

, as discussed in §18.2(xi). …

3: 3.8 Nonlinear Equations

… ►However, to guard against the accumulation of rounding errors, a final iteration for each zero should also be performed on the original polynomial

p(z)

. …

4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In unusual cases

N=\infty

, even for all

\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). …