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1: 31.10 Integral Equations and Representations

… ►and the kernel $𝒦(z,t)$ is a solution of the partial differential equation … ►

Kernel Functions

… ►and the kernel $𝒦(z;s,t)$ is a solution of the partial differential equation … ►

Kernel Functions

… ►leads to the kernel equation …

2: 28.10 Integral Equations

… ►

§28.10(i) Equations with Elementary Kernels

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§28.10(ii) Equations with Bessel-Function Kernels

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3: 10.67 Asymptotic Expansions for Large Argument

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

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

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10.67.13 $${\ker }^{2}x+{\mathrm{kei}}^{2}x\sim \frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(1-\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^2}+\frac{33}{256\sqrt{2}}\frac{1}{x^3}-\frac{1797}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

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:

9.10.31

Permalink:

http://dlmf.nist.gov/10.67.E13

Encodings:

pMML, png, TeX

See also:

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

►

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

ⓘ

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:

pMML, png, TeX

See also:

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

►

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

ⓘ

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

►

10.67.16 $${\left({\ker }^{\prime }x\right)}^2+{\left({\mathrm{kei}}^{\prime }x\right)}^2\sim \frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(1+\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9}{64}\frac{1}{x^2}-\frac{75}{256\sqrt{2}}\frac{1}{x^3}+\frac{2475}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right).$$

ⓘ

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:

9.10.34

Permalink:

http://dlmf.nist.gov/10.67.E16

Encodings:

pMML, png, TeX

See also:

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

4: 10.69 Uniform Asymptotic Expansions for Large Order

… ►

10.69.3 $${\ker }_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }\left(\nu x\right)\sim {\mathrm{e}}^{-\nu \xi }{\left(\frac{\pi }{2\nu \xi }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

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, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.38

Permalink:

http://dlmf.nist.gov/10.69.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

… ►

10.69.5 $${\ker }_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }^{\prime }\left(\nu x\right)\sim -\frac{{\mathrm{e}}^{-\nu \xi }}{x}{\left(\frac{\pi \xi }{2\nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

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, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.40

Permalink:

http://dlmf.nist.gov/10.69.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

…

5: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

§28.28(i) Equations with Elementary Kernels

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6: 10.61 Definitions and Basic Properties

… ►

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

ⓘ

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

… ►

10.61.3 $$x^2\frac{d^{2}w}{{dx}^2}+x\frac{dw}{dx}-(\mathrm{i}{x}^2+{\nu }^2)w=0,$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{i}$: imaginary unit, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.3

Referenced by:

§10.61(ii), §10.68(ii)

Permalink:

http://dlmf.nist.gov/10.61.E3

Encodings:

pMML, png, TeX

See also:

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

►

10.61.4 $$x^4\frac{d^{4}w}{{dx}^4}+2x^3\frac{d^{3}w}{{dx}^3}-(1+2{\nu }^2)\left(x^2\frac{d^{2}w}{{dx}^2}-x\frac{dw}{dx}\right)+({\nu }^4-4{\nu }^2+x^4)w=0,$$ $w={\mathrm{ber}}_{\pm \nu }x,{\mathrm{bei}}_{\pm \nu }x,{\ker }_{\pm \nu }x,{\mathrm{kei}}_{\pm \nu }x$.

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.4

Referenced by:

§10.61(ii)

Permalink:

http://dlmf.nist.gov/10.61.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

10.61.11 $${\ker }_{\frac{1}{2}}\left(x\sqrt{2}\right)={\mathrm{kei}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi }{x}}{\mathrm{e}}^{-x}\sin \left(x-\frac{\pi }{8}\right),$$

ⓘ

Symbols:

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

Referenced by:

§10.61(v)

Permalink:

http://dlmf.nist.gov/10.61.E11

Encodings:

pMML, png, TeX

See also:

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

►

10.61.12 $${\mathrm{kei}}_{\frac{1}{2}}\left(x\sqrt{2}\right)=-{\ker }_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi }{x}}{\mathrm{e}}^{-x}\cos \left(x-\frac{\pi }{8}\right).$$

ⓘ

Symbols:

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

Referenced by:

§10.61(v), Ch.10

Permalink:

http://dlmf.nist.gov/10.61.E12

Encodings:

pMML, png, TeX

See also:

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

7: 10.65 Power Series

… ►

10.65.3 $${\ker }_{n}x=\frac{1}{2}{(\frac{1}{2}x)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{ber}}_{n}x+\frac{1}{4}\pi {\mathrm{bei}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k,$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $x$: real variable

A&S Ref:

9.9.11

Permalink:

http://dlmf.nist.gov/10.65.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(ii), §10.65 and Ch.10

…

8: 1.15 Summability Methods

… ►

1.15.12 $$P(r,\theta )=\frac{1-r^2}{1-2r\cos \theta +r^2}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{r}^{\left|n\right|}{\mathrm{e}}^{\mathrm{i}n\theta },$$ ,

ⓘ

Defines:

$P(r,\theta )$: Poisson kernel (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

… ►

1.15.14 $$P(r,\theta )\to 0,$$

ⓘ

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

… ►

1.15.15 $$K_n(\theta )=\frac{1}{n+1}{\left(\frac{\sin \left(\frac{1}{2}(n+1)\theta \right)}{\sin \left(\frac{1}{2}\theta \right)}\right)}^2,$$

ⓘ

Defines:

$K_n(\theta )$: Fejér kernel (locally)

Symbols:

$\sin z$: sine function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.15.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

… ►

1.15.17 $$K_n(\theta )\to 0,$$

ⓘ

Symbols:

$n$: nonnegative integer and $K_n(\theta )$: Fejér kernel

Permalink:

http://dlmf.nist.gov/1.15.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

… ►

1.15.42 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{K}_R(s)ds=1.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral and $K_R(s)$: Fejér kernel

Permalink:

http://dlmf.nist.gov/1.15.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

…

9: 10.68 Modulus and Phase Functions

… ►

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

ⓘ

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

… ►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. …

10: 18.2 General Orthogonal Polynomials

… ►

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

ⓘ

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

… ►

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

ⓘ

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

…