Stieltjes

SearchAdvancedHelp

(0.001 seconds)

11—20 of 30 matching pages

11: Bibliography R

… ►

Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

ⓘ

Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

►

Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

ⓘ

External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

►

Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

ⓘ

External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

…

12: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. …

13: 25.6 Integer Arguments

… ►

25.6.12 $${\zeta }^{\prime \prime }\left(0\right)=-\frac{1}{2}{(\ln \left(2\pi \right))}^2+\frac{1}{2}{\gamma }^2-\frac{1}{24}{\pi }^2+{\gamma }_1,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: principal branch of logarithm function

Source:

Apostol (1985a, pp. 226, 227, 229)

Referenced by:

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

Permalink:

http://dlmf.nist.gov/25.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

►where ${\gamma }_1$ is given by (25.2.5). …

14: 9.10 Integrals

… ►

§9.10(vii) Stieltjes Transforms

…

15: 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

ⓘ

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

… ►

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

ⓘ

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

►

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

ⓘ

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

… ►

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

ⓘ

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

…

16: Bibliography W

… ►

Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

ⓘ

External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

… ►

Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions. Proc. Roy. Soc. London Ser. A 458, pp. 625–644.

ⓘ

External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

…

17: 18.1 Notation

… ►

Stieltjes–Wigert: $S_n(x;q)$.

…

18: Bibliography J

… ►

Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

ⓘ

External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

BibTeX

…

19: Bibliography N

… ►

On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

ⓘ

External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

…

20: Bibliography V

… ►

Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

ⓘ

External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

Encodings:

BibTeX

…