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11: 10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

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10.32.11 $$K_{\nu }\left(xz\right)=\frac{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right){(2z)}^{\nu }}{{\pi }^{\frac{1}{2}}{x}^{\nu }}{\int }_0^{\mathrm{\infty }}\frac{\cos \left(xt\right)dt}{{(t^2+z^2)}^{\nu +\frac{1}{2}}},$$ $\mathrm{\Re }\nu >-\frac{1}{2}$, $x>0$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.25 (corrected)

Permalink:

http://dlmf.nist.gov/10.32.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32(i), §10.32 and Ch.10

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12: 12.5 Integral Representations

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§12.5(i) Integrals Along the Real Line

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13: 19.33 Triaxial Ellipsoids

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19.33.10 $$U=\frac{1}{2}{\int }_{{ℝ}^6}\frac{\rho (x,y,z)\rho (x^{\prime },y^{\prime },z^{\prime })dxdydzd{x}^{\prime }d{y}^{\prime }d{z}^{\prime }}{\sqrt{{(x-x^{\prime })}^2+{(y-y^{\prime })}^2+{(z-z^{\prime })}^2}}.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $ℝ$: real line, $\rho (x,y,z)$: density and $U$: energy

Permalink:

http://dlmf.nist.gov/19.33.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(iv), §19.33 and Ch.19

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14: 1.16 Distributions

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1.16.26 $$⟨{\mathrm{\Lambda }}_f,\varphi ⟩={\int }_{{ℝ}^n}f(x)\varphi (x)dx,$$ $\varphi \in {𝒟}_n$.

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Symbols:

$dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral, $ℝ$: real line, $n$: nonnegative integer, ${𝒟}_n$: space of test functions, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function and $\mathrm{\Lambda }$: mapping

Permalink:

http://dlmf.nist.gov/1.16.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(vi), §1.16 and Ch.1

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1.16.27 $$⟨{𝐷}^{k}f,\varphi ⟩={(-1)}^{\left|k\right|}{\int }_{{ℝ}^n}f(x){\varphi }^{(k)}(x)dx,$$ $\varphi \in {𝒟}_n$,

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Symbols:

$dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral, $ℝ$: real line, $k$: integer, $n$: nonnegative integer, ${𝒟}_n$: space of test functions, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.16.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(vi), §1.16 and Ch.1

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1.16.29 $$ℱ(\varphi )(𝐱)=ℱ\varphi (𝐱)=\frac{1}{{(2\pi )}^{n/2}}{\int }_{{ℝ}^n}\varphi (𝐭){\mathrm{e}}^{\mathrm{i}𝐱\cdot 𝐭}d𝐭,$$

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Defines:

$ℱ(\varphi )$: Fourier transform of a test function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line, $n$: nonnegative integer and $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function

Referenced by:

(1.14.1), (1.16.33), (1.16.34), (1.16.35), item (i), item (i)

Permalink:

http://dlmf.nist.gov/1.16.E29

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$F(𝐱)=F=\frac{1}{{(2\pi )}^{n/2}}{\int }_{{ℝ}^n}\varphi (𝐭){\mathrm{e}}^{\mathrm{i}𝐱\cdot 𝐭}d𝐭.$$

The notation used in §1.14(i) is extended to a test function of several variables.

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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1.16.33 $$ℱ(P(𝐃)\varphi )(𝐱)=P(-𝐱)ℱ\varphi (𝐱),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line, $n$: nonnegative integer, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $ℱ(\varphi )$: Fourier transform of a test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E33

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$\frac{1}{{(2\pi )}^{n/2}}{\int }_{{ℝ}^n}(P(𝐷)\varphi )(𝐭){\mathrm{e}}^{\mathrm{i}𝐱\cdot 𝐭}d𝐭=P(-𝐱)F(𝐱).$$

It has been rewritten using the notation of (1.16.29) and (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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1.16.34 $$ℱ(P\varphi )(𝐱)=P(𝐃)ℱ\varphi (𝐱).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line, $n$: nonnegative integer, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $ℱ(\varphi )$: Fourier transform of a test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E34

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$\frac{1}{{(2\pi )}^{n/2}}{\int }_{{ℝ}^n}P(𝐭)\varphi (𝐭){\mathrm{e}}^{\mathrm{i}𝐱\cdot 𝐭}d𝐭=P(𝐷)F(𝐱).$$

It has been rewritten using the notation of (1.16.29) and (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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15: 10.9 Integral Representations

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§10.9(i) Integrals along the Real Line

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10.9.11 $$H_{\nu }^{(2)}\left(z\right)=-\frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}z\cosh t-\nu t}dt,$$ .

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Symbols:

$H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.9(i)

Permalink:

http://dlmf.nist.gov/10.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

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10.9.16 $${\left(\frac{z+\zeta }{z-\zeta }\right)}^{\frac{1}{2}\nu }{H}_{\nu }^{(2)}\left({(z^2-{\zeta }^2)}^{\frac{1}{2}}\right)=-\frac{1}{\pi \mathrm{i}}{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}z\cosh t-\mathrm{i}\zeta \sinh t-\nu t}dt,$$ .

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Symbols:

$H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.9(i)

Permalink:

http://dlmf.nist.gov/10.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

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16: Bibliography I

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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17: 1.17 Integral and Series Representations of the Dirac Delta

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1.17.2 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(x-a\right)\varphi (x)dx=\varphi (a),$$ $a\in ℝ$,

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line and $\varphi (x)$: continuous function

Referenced by:

§1.16(viii), §1.17(i), §1.17(ii)

Permalink:

http://dlmf.nist.gov/1.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(i), §1.17 and Ch.1

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18: 29.12 Definitions

… ►The superscript $m$ on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of $z$-zeros of each Lamé polynomial in the interval $(0,K)$, while $n-m$ is the number of $z$-zeros in the open line segment from $K$ to $K+\mathrm{i}{K}^{\prime }$. …

19: 25.5 Integral Representations

§25.5 Integral Representations

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25.5.19 $$\zeta \left(m+s\right)={(-1)}^{m-1}\frac{\mathrm{\Gamma }\left(s\right)\sin \left(\pi s\right)}{\pi \mathrm{\Gamma }\left(m+s\right)}{\int }_0^{\mathrm{\infty }}{\psi }^{(m)}\left(1+x\right)x^{-s}dx,$$ $m=1,2,3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, p. 178)

Referenced by:

§25.5(ii)

Permalink:

http://dlmf.nist.gov/25.5.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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20: 9.12 Scorer Functions

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9.12.19 $$\mathrm{Gi}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\sin \left(\frac{1}{3}{t}^3+xt\right)dt,$$ $x\in ℝ$.

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line, $\sin z$: sine function and $x$: real variable

Proof sketch:

Combine (9.5.3) and (9.12.11).

Permalink:

http://dlmf.nist.gov/9.12.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

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