Poisson kernels

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1: 1.15 Summability Methods

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Poisson Kernel

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1.15.13 $$\frac{1}{2\pi }{\int }_0^{2\pi }P(r,\theta )d\theta =1.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $P(r,\theta )$: Poisson kernel

Permalink:

http://dlmf.nist.gov/1.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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1.15.14 $$P(r,\theta )\to 0,$$

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Symbols:

$P(r,\theta )$: Poisson kernel

Permalink:

http://dlmf.nist.gov/1.15.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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1.15.20 $$A(r,\theta )=\frac{1}{2\pi }{\int }_0^{2\pi }P(r,\theta -t)f(t)dt.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $P(r,\theta )$: Poisson kernel and $A(r,\theta )$: Abel mean

Permalink:

http://dlmf.nist.gov/1.15.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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Poisson Kernel

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2: 18.18 Sums

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§18.18(vii) Poisson Kernels

►See (18.2.41) for the Poisson kernel in case of general OP’s. ►

Laguerre

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Hermite

… ►For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938). …

3: 18.2 General Orthogonal Polynomials

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Poisson kernel

►For OP’s $p_n$ with $h_n$ and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by …Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12). …

4: 18.12 Generating Functions

… ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

5: Bibliography T

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

Encodings:

BibTeX

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On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.

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External Links:

ISSN 0378-4371, Document, MathReview (Vitor R. Vieira)

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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