Stieltjes with jumps

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3 matching pages

1: 1.4 Calculus of One Variable

… ►The utility of the generalization implicit in the Stieltjes measure appears when

\alpha(x)

is not everywhere continuous, but has discontinuous jumps at specific values of

x

, say

x_{n}\in(a,b)

. See Riesz and Sz.-Nagy (1990, Ch. 3). … ►

1.4.23_3

\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=\int_{a}^{b}w(x)f(x)\,\mathrm{d}x+\sum_{% n=1}^{N}w_{n}f(x_{n}).

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $w(x)$ : weight function
Referenced by:: §1.4(v), Erratum (V1.2.0) §1.4
Permalink:: http://dlmf.nist.gov/1.4.E23_3
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

…

2: 18.39 Applications in the Physical Sciences

… ►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{-}

. …

3: 18.40 Methods of Computation

… ►Interpolation of the midpoints of the jumps followed by differentiation with respect to

x

yields a Stieltjes–Perron inversion to obtain

w^{\mathrm{RCP}}(x)

to a precision of

\sim 4

decimal digits for

N=120

. …