Poisson integral

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1: 1.15 Summability Methods

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Poisson Kernel

… ►is the Poisson integral of $f(t)$. … ►is the conjugate Poisson integral of $f(x)$. … …

2: 1.9 Calculus of a Complex Variable

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Poisson Integral

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3: 10.9 Integral Representations

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Poisson’s and Related Integrals

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4: 19.18 Derivatives and Differential Equations

… ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). ►The function $w=R_{-a}(\frac{1}{2},\frac{1}{2};x+y,x-y)$ satisfies an Euler–Poisson–Darboux equation: …

5: Bibliography K

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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).

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Notes:

In cooperation with W. Rudert

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MathReview, ZentralBlatt

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1st item.

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6: Bibliography H

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Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

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§17.13.

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Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

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External Links:

MathReview (C. Tanaka)

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§7.14(iii).

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Elliptic Integrals. Dover Publications Inc., New York.

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Notes:

Unaltered reprint of original edition published by Wiley, New York, 1917.

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MathReview Entry, ZentralBlatt

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§19.8(ii).

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Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.

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Notes:

Double-precision Fortran, maximum accuracy 12D.

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FullText, Code

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5th item.

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Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.

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ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt

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§14.31(i).

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7: 1.8 Fourier Series

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1.8.4 $$c_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x){\mathrm{e}}^{-\mathrm{i}nx}dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.17(iii)

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http://dlmf.nist.gov/1.8.E4

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Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.5 $$\frac{1}{2}{a}_0^2+\sum _{n=1}^{\mathrm{\infty }}(a_n^2+b_n^2)\le \frac{1}{\pi }{\int }_{-\pi }^{\pi }{(f(x))}^{2}dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

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http://dlmf.nist.gov/1.8.E5

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Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►

Poisson’s Summation Formula

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1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

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http://dlmf.nist.gov/1.8.E16

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Annotations for §1.8(iv), §1.8(iv), §1.8 and Ch.1

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8: Bibliography B

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A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

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§12.12.

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Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.

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Document, ZentralBlatt

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§19.2(iii).

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Normalization integrals of orthogonal Heun functions. J. Math. Phys. 38 (7), pp. 3692–3699.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§31.9(i).

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Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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External Links:

Document, MathReview (T. M. Apostol), ZentralBlatt

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§24.16(ii).

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On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

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ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.13(i).

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9: 3.5 Quadrature

… ►See also Poisson’s summation formula (§1.8(iv)). … ►

§3.5(vii) Oscillatory Integrals

►Integrals of the form … ► … ►For the Bromwich integral …

10: 14.31 Other Applications

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§14.31(i) Toroidal Functions

►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …