beij%C3%for%20kernel

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1: Qiming Wang

… ► 1943 in Shanghai, China) received a degree in Computational Mathematics from Tsinghua University, Beijing, in 1966. …

2: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

3: 10.75 Tables

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Achenbach (1986) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0(.1)8$, 20D or 18–20S.

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

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Zhang and Jin (1996, p. 322) tabulates $\mathrm{ber}x$, ${\mathrm{ber}}^{\prime }x$, $\mathrm{bei}x$, ${\mathrm{bei}}^{\prime }x$, $\ker x$, ${\ker }^{\prime }x$, $\mathrm{kei}x$, ${\mathrm{kei}}^{\prime }x$, $x=0(1)20$, 7S.

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Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\mathrm{ber}x$, ${\mathrm{ber}}^{\prime }x$, $\mathrm{bei}x$, ${\mathrm{bei}}^{\prime }x$, $\ker x$, ${\ker }^{\prime }x$, $\mathrm{kei}x$, ${\mathrm{kei}}^{\prime }x$, 8D.

4: 20 Theta Functions

Chapter 20 Theta Functions

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5: 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

6: 1.15 Summability Methods

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Poisson Kernel

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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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Fejér Kernel

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Poisson Kernel

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Fejér Kernel

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7: 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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8: 8 Incomplete Gamma and Related
Functions

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9: 28 Mathieu Functions and Hill’s Equation

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10: 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 …