with Bessel-function kernels

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1: 28.10 Integral Equations

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§28.10(ii) Equations with Bessel-Function Kernels

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2: 18.18 Sums

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Laguerre

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3: 37.12 Orthogonal Polynomials on Quadratic Surfaces

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4: 37.18 Orthogonal Polynomials on Quadratic Domains

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5: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. …

6: 10.62 Graphs

§10.62 Graphs

►For the modulus functions $M\left(x\right)$ and $N\left(x\right)$ see §10.68(i) with $\nu =0$. … ►

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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 10.62.3: ${\mathrm{e}}^{-x/\sqrt{2}}\mathrm{ber}x$, ${\mathrm{e}}^{-x/\sqrt{2}}\mathrm{bei}x$, ${\mathrm{e}}^{-x/\sqrt{2}}M\left(x\right)$, $0\le x\le 8$. 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

7: 37.16 Orthogonal Polynomials on the Hyperoctant

… ►define the weight function … ►

§37.16(ii) Poisson Kernel

►The Poisson kernel (37.13.6) of ${𝒱}_n^{𝜶}({ℝ}_{+}^d)$ is given explicitly by …For the modified Bessel function $I_{\nu }\left(z\right)$ see (10.25.2). …

8: 10.68 Modulus and Phase Functions

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10.68.2 $$N_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\varphi }_{\nu }\left(x\right)}={\ker }_{\nu }x+\mathrm{i}{\mathrm{kei}}_{\nu }x,$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.68.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(i), §10.68 and Ch.10

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9: 10.61 Definitions and Basic Properties

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10.61.2 $${\ker }_{\nu }x+\mathrm{i}{\mathrm{kei}}_{\nu }x={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(x{\mathrm{e}}^{\pi \mathrm{i}/4}\right)=\frac{1}{2}\pi \mathrm{i}{H}_{\nu }^{(1)}\left(x{\mathrm{e}}^{3\pi \mathrm{i}/4}\right)=-\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{-\nu \pi \mathrm{i}}{H}_{\nu }^{(2)}\left(x{\mathrm{e}}^{-\pi \mathrm{i}/4}\right).$$

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Defines:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function and ${\ker }_{\nu }\left(x\right)$: Kelvin function

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.2

Referenced by:

§10.63(ii), §10.65(ii), §10.67(i), §10.69

Permalink:

http://dlmf.nist.gov/10.61.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.61(i), §10.61 and Ch.10

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10: 10.71 Integrals

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§10.71(i) Indefinite Integrals

… ►where $M_{\nu }\left(x\right)$ and $N_{\nu }\left(x\right)$ are the modulus functions introduced in §10.68(i). ►

§10.71(ii) Definite Integrals

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§10.71(iii) Compendia

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