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1: 31.10 Integral Equations and Representations

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Kernel Functions

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31.10.8 $${\sin }^2\theta \left(\frac{{\partial }^2𝒦}{{\partial \theta }^2}+\left((1-2\gamma )\tan \theta +2(\delta +ϵ-\frac{1}{2})\cot \theta \right)\frac{\partial 𝒦}{\partial \theta }-4\alpha \beta 𝒦\right)+\frac{{\partial }^2𝒦}{{\partial \varphi }^2}+\left((1-2\delta )\cot \varphi -(1-2ϵ)\tan \varphi \right)\frac{\partial 𝒦}{\partial \varphi }=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel, $\theta$: angle and $\varphi$: angle

Referenced by:

§31.10(i)

Permalink:

http://dlmf.nist.gov/31.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

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Kernel Functions

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31.10.18 $$\frac{{\partial }^2𝒦}{{\partial u}^2}+\frac{{\partial }^2𝒦}{{\partial v}^2}+\frac{{\partial }^2𝒦}{{\partial w}^2}+\frac{2\gamma -1}{u}\frac{\partial 𝒦}{\partial u}+\frac{2\delta -1}{v}\frac{\partial 𝒦}{\partial v}+\frac{2ϵ-1}{w}\frac{\partial 𝒦}{\partial w}=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝒦(z;s,t)$: kernel, $u$ , $v$ and $w$

Permalink:

http://dlmf.nist.gov/31.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

… ►leads to the kernel equation …

2: 28.10 Integral Equations

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§28.10(i) Equations with Elementary Kernels

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§28.10(ii) Equations with Bessel-Function Kernels

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3: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

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4: 18.2 General Orthogonal Polynomials

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18.2.12 $$K_n(x,y)\equiv \sum _{\mathrm{\ell }=0}^n\frac{p_{\mathrm{\ell }}(x)p_{\mathrm{\ell }}(y)}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y},$$ $x\ne y$,

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Symbols:

$\equiv$: equals by definition, $(a,b)$: open interval, $y$: real variable, $p_n(x)$: polynomial of degree $n$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $h_n$ and $k_n$

A&S Ref:

22.12.1

Referenced by:

§18.2(v), Erratum (V1.2.0) for Equations (18.2.12), (18.2.13)

Permalink:

http://dlmf.nist.gov/18.2.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

The left-hand side was updated to include the definition of the Christoffel–Darboux kernel $K_n(x,y)$.

See also:

Annotations for §18.2(v), §18.2 and Ch.18

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18.2.13 $$K_n(x,x)=\sum _{\mathrm{\ell }=0}^n\frac{{(p_{\mathrm{\ell }}(x))}^2}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\left(p_{n+1}^{\prime }(x)p_n(x)-p_n^{\prime }(x)p_{n+1}(x)\right).$$

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Symbols:

$(a,b)$: open interval, $p_n(x)$: polynomial of degree $n$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $h_n$ and $k_n$

Referenced by:

Erratum (V1.2.0) for Equations (18.2.12), (18.2.13)

Permalink:

http://dlmf.nist.gov/18.2.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

The left-hand side was updated to include the definition of the confluent form of the Christoffel–Darboux kernel $K_n(x,x)$.

See also:

Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

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5: 28.32 Mathematical Applications

… ►Kernels $K$ can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. …

6: 10.63 Recurrence Relations and Derivatives

… ►Equations (10.63.6) and (10.63.7) also hold when the symbols $\mathrm{ber}$ and $\mathrm{bei}$ in (10.63.5) are replaced throughout by $\ker$ and $\mathrm{kei}$, respectively. …

7: 10.68 Modulus and Phase Functions

… ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. …

8: Errata

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18.2.12 $$K_n(x,y)\equiv \sum _{\mathrm{\ell }=0}^n\frac{p_{\mathrm{\ell }}(x)p_{\mathrm{\ell }}(y)}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y},$$ $x\ne y$

18.2.13 $$K_n(x,x)=\sum _{\mathrm{\ell }=0}^n\frac{{(p_{\mathrm{\ell }}(x))}^2}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\left(p_{n+1}^{\prime }(x)p_n(x)-p_n^{\prime }(x)p_{n+1}(x)\right)$$

The left-hand sides were updated to include the definition of the Christoffel–Darboux kernel $K_n(x,y)$.

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9: 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. ►

§10.61(ii) Differential Equations

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

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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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10: 13.27 Mathematical Applications

… ►The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).