# Stieltjes with jumps

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3 matching pages ♦

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## 3 matching pages

##### 1: 1.4 Calculus of One Variable

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►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
$${\int}_{a}^{b}f(x)d\alpha (x)={\int}_{a}^{b}w(x)f(x)dx+\sum _{n=1}^{N}{w}_{n}f({x}_{n}).$$

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##### 2: 18.39 Applications in the Physical Sciences

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►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 $x\to -1-$.
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##### 3: 18.40 Methods of Computation

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►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 $N=120$.
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