Poisson%20kernel

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11: 1.8 Fourier Series

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§1.8(iv) 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)

Permalink:

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

Encodings:

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See also:

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

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12: Bibliography H

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Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35 (2), pp. 405–433.

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External Links:

Document, MathReview (E. L. Koh), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

§14.31(i).

Encodings:

BibTeX

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13: 19.18 Derivatives and Differential Equations

… ►and also a system of $n(n-1)/2$ Euler–Poisson differential equations (of which only $n-1$ are independent): … ►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: …

14: 18.12 Generating Functions

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

15: 10.62 Graphs

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Figure 10.62.2: $\ker x$, $\mathrm{kei}x$, ${\ker }^{\prime }x$, ${\mathrm{kei}}^{\prime }x$, $0\le x\le 8$. Magnify

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Figure 10.62.4: ${\mathrm{e}}^{x/\sqrt{2}}\ker x$, ${\mathrm{e}}^{x/\sqrt{2}}\mathrm{kei}x$, ${\mathrm{e}}^{x/\sqrt{2}}N\left(x\right)$, $0\le x\le 8$. Magnify

16: 10.61 Definitions and Basic Properties

… ►When $\nu =0$ suffices on $\mathrm{ber}$, $\mathrm{bei}$, $\ker$, and $\mathrm{kei}$ are usually suppressed. ►Most properties of ${\mathrm{ber}}_{\nu }x$, ${\mathrm{bei}}_{\nu }x$, ${\ker }_{\nu }x$, and ${\mathrm{kei}}_{\nu }x$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. … ►

$${\ker }_{-\nu }x=\cos \left(\nu \pi \right){\ker }_{\nu }x-\sin \left(\nu \pi \right){\mathrm{kei}}_{\nu }x,$$

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$${\mathrm{kei}}_{-\nu }x=\sin \left(\nu \pi \right){\ker }_{\nu }x+\cos \left(\nu \pi \right){\mathrm{kei}}_{\nu }x.$$

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$${\ker }_{-n}x={(-1)}^n{\ker }_{n}x,{\mathrm{kei}}_{-n}x={(-1)}^n{\mathrm{kei}}_{n}x.$$

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17: 8 Incomplete Gamma and Related
Functions

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18: 28 Mathieu Functions and Hill’s Equation

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19: 31.10 Integral Equations and Representations

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Kernel Functions

… ►The kernel $𝒦$ must satisfy … ►

Kernel Functions

… ►The kernel $𝒦$ must satisfy … ►leads to the kernel equation …

20: 10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

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10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

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See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). … ►

10.67.14 $$\ker x{\mathrm{kei}}^{\prime }x-{\ker }^{\prime }x\mathrm{kei}x\sim -\frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}-\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^2}-\frac{39}{512}\frac{1}{x^3}+\frac{75}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.32

Permalink:

http://dlmf.nist.gov/10.67.E14

Encodings:

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See also:

Annotations for §10.67(ii), §10.67 and Ch.10

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10.67.15 $$\ker x{\ker }^{\prime }x+\mathrm{kei}x{\mathrm{kei}}^{\prime }x\sim -\frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^2}+\frac{45}{512}\frac{1}{x^3}+\frac{315}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.33

Permalink:

http://dlmf.nist.gov/10.67.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

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