Stieltjes electrostatic interpretation

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1: 31.15 Stieltjes Polynomials

§31.15 Stieltjes Polynomials

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§31.15(ii) Zeros

… ►This is the Stieltjes electrostatic interpretation. … ►

§31.15(iii) Products of Stieltjes Polynomials

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2: 2.6 Distributional Methods

… ►With this interpretation … ►

§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of $f(t)$ is defined by … ►Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …

3: 18.39 Applications in the Physical Sciences

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. … 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 $x\to -1-$. … ►The equivalent quadrature weight, $w_i/w^{\mathrm{CP}}(x_i)$, also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii). … ►For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).

4: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

… ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain $w(x)$. ►

Stieltjes Inversion via (approximate) Analytic Continuation

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Histogram Approach

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Derivative Rule Approach

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5: 29.12 Definitions

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29.12.13 $$\frac{\rho +\frac{1}{4}}{{\xi }_p}+\frac{\sigma +\frac{1}{4}}{{\xi }_p-1}+\frac{\tau +\frac{1}{4}}{{\xi }_p-k^{-2}}+\sum _{\begin{array}{c}q=1\\ q\ne p\end{array}}^n\frac{1}{{\xi }_p-{\xi }_q}=0,$$ $p=1,2,\mathrm{\dots },n$.

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Symbols:

$n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, ($0$ or $\frac{1}{2}$): parameter $\rho$ , ($0$ or $\frac{1}{2}$): parameter $\sigma$ , ($0$ or $\frac{1}{2}$): parameter $\tau$ and ${\xi }_n$: zeros

Referenced by:

§29.20(iii)

Permalink:

http://dlmf.nist.gov/29.12.E13

Encodings:

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See also:

Annotations for §29.12(iii), §29.12 and Ch.29

►This result admits the following electrostatic interpretation: Given three point masses fixed at $t=0$, $t=1$, and $t=k^{-2}$ with positive charges $\rho +\frac{1}{4}$, $\sigma +\frac{1}{4}$, and $\tau +\frac{1}{4}$, respectively, and $n$ movable point masses at $t_1,t_2,\mathrm{\dots },t_n$ arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when $t_j={\xi }_j$ for $j=1,2,\mathrm{\dots },n$.

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For a Lebesgue–Stieltjes measure $d\alpha$ on $X$ let $L^2(X,d\alpha )$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $d\alpha$, ►

1.18.11

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.13 $$c_n=⟨f,{\varphi }_n⟩={\int }_a^{b}f(x)\overline{{\varphi }_n(x)}dx,$$ $f\in L^2\left(X\right)$,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.14 $${\int }_a^b{\left|f(x)\right|}^{2}dx=\sum _{n=0}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$ $f\in L^2\left(X\right)$,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.64 $$f(x)={\int }_{{𝝈}_c}\widehat{f}(\lambda ){\varphi }_{\lambda }(x)d\lambda +\sum _{{𝝈}_p}\widehat{f}({\lambda }_n){\varphi }_{{\lambda }_n}(x),$$ $f(x)\in C(X)\cap L^2\left(X\right)$.

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $\cap$: intersection, $C(I)$ or $C(a,b)$: continuous on an interval $I$ or $(a,b)$ and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.18.E64

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(viii), §1.18 and Ch.1

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7: 25.2 Definition and Expansions

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25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

►where the Stieltjes constants ${\gamma }_n$ are defined via ►

25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

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Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

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See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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8: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

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18.29.2 $$Q_n(z;a,b,c,d\mid q)\sim \frac{z^n{(a{z}^{-1},b{z}^{-1},c{z}^{-1},d{z}^{-1};q)}_{\mathrm{\infty }}}{{(z^{-2},bc,bd,cd;q)}_{\mathrm{\infty }}},$$ $n\to \mathrm{\infty }$; $z,a,b,c,d,q$ fixed.

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Symbols:

$\sim$: asymptotic equality, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.29.E2

Encodings:

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See also:

Annotations for §18.29 and Ch.18

►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …

9: Bibliography K

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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.

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External Links:

Document

Referenced by:

§32.17.

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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10: 1.4 Calculus of One Variable

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Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals

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