# §1.16 Distributions

## §1.16(i) Test Functions

Let $\phi$ be a function defined on an open interval $I=(a,b)$, which can be infinite. The closure of the set of points where $\phi\not=0$ is called the support of $\phi$. If the support of $\phi$ is a compact set (§1.9(vii)), then $\phi$ is called a function of compact support. A test function is an infinitely differentiable function of compact support.

A sequence $\{\phi_{n}\}$ of test functions converges to a test function $\phi$ if the support of every $\phi_{n}$ is contained in a fixed compact set $K$ and as $n\to\infty$ the sequence $\{\phi_{n}^{(k)}\}$ converges uniformly on $K$ to $\phi^{(k)}$ for $k=0,1,2,\dots$.

The linear space of all test functions with the above definition of convergence is called a test function space. We denote it by $\mathcal{D}(I)$.

A mapping $\Lambda$ on $\mathcal{D}(I)$ is a linear functional if it takes complex values and

 1.16.1 $\Lambda(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2})=\alpha_{1}\Lambda(\phi_{1})+% \alpha_{2}\Lambda(\phi_{2}),$ ⓘ Symbols: $\phi(x)$: test function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E1 Encodings: TeX, pMML, png See also: Annotations for §1.16(i), §1.16 and Ch.1

where $\alpha_{1}$ and $\alpha_{2}$ are real or complex constants. $\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}$ is called a distribution if it is a continuous linear functional on $\mathcal{D}(I)$, that is, it is a linear functional and for every $\phi_{n}\to\phi$ in $\mathcal{D}(I)$,

 1.16.2 $\lim_{n\to\infty}\Lambda(\phi_{n})=\Lambda(\phi).$ ⓘ Symbols: $n$: nonnegative integer, $\phi(x)$: test function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E2 Encodings: TeX, pMML, png See also: Annotations for §1.16(i), §1.16 and Ch.1

From here on we write $\left\langle\Lambda,\phi\right\rangle$ for $\Lambda(\phi)$. The space of all distributions will be denoted by $\mathcal{D}^{*}(I)$. A distribution $\Lambda$ is called regular if there is a function $f$ on $I$, which is absolutely integrable on every compact subset of $I$, such that

 1.16.3 $\left\langle\Lambda,\phi\right\rangle=\int_{I}f(x)\phi(x)\mathrm{d}x.$

We denote a regular distribution by $\Lambda_{f}$, or simply $f$, where $f$ is the function giving rise to the distribution. (If a distribution is not regular, it is called singular.)

Define

 1.16.4 $\left\langle\Lambda_{1}+\Lambda_{2},\phi\right\rangle=\left\langle\Lambda_{1},% \phi\right\rangle+\left\langle\Lambda_{2},\phi\right\rangle,$ ⓘ Symbols: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $\phi(x)$: test function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E4 Encodings: TeX, pMML, png See also: Annotations for §1.16(i), §1.16 and Ch.1
 1.16.5 $\left\langle c\Lambda,\phi\right\rangle=c\left\langle\Lambda,\phi\right\rangle% =\left\langle\Lambda,c\phi\right\rangle,$ ⓘ Symbols: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $\phi(x)$: test function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E5 Encodings: TeX, pMML, png See also: Annotations for §1.16(i), §1.16 and Ch.1

where $c$ is a constant. More generally, if $\alpha(x)$ is an infinitely differentiable function, then

 1.16.6 $\left\langle\alpha\Lambda,\phi\right\rangle=\left\langle\Lambda,\alpha\phi% \right\rangle.$ ⓘ Symbols: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $\phi(x)$: test function and $\Lambda$: mapping Referenced by: §1.16(vii) Permalink: http://dlmf.nist.gov/1.16.E6 Encodings: TeX, pMML, png See also: Annotations for §1.16(i), §1.16 and Ch.1

We say that a sequence of distributions $\{\Lambda_{n}\}$ converges to a distribution $\Lambda$ in $\mathcal{D}^{*}$ if

 1.16.7 $\lim_{n\to\infty}\left\langle\Lambda_{n},\phi\right\rangle=\left\langle\Lambda% ,\phi\right\rangle$

for all $\phi\in\mathcal{D}(I)$.

## §1.16(ii) Derivatives of a Distribution

The derivative $\Lambda^{\prime}$ of a distribution is defined by

 1.16.8 $\left\langle\Lambda^{\prime},\phi\right\rangle=-\left\langle\Lambda,\phi^{% \prime}\right\rangle,$ $\phi\in\mathcal{D}(I)$.

Similarly

 1.16.9 $\left\langle\Lambda^{(k)},\phi\right\rangle=(-1)^{k}\left\langle\Lambda,\phi^{% (k)}\right\rangle,$ $k=1,2,\dots$. ⓘ Symbols: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $k$: integer, $\phi(x)$: test function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E9 Encodings: TeX, pMML, png See also: Annotations for §1.16(ii), §1.16 and Ch.1

For any locally integrable function $f$, its distributional derivative is $Df=\Lambda^{\prime}_{f}$.

## §1.16(iii) Dirac Delta Distribution

 1.16.10 $\displaystyle\left\langle\delta,\phi\right\rangle$ $\displaystyle=\phi(0),$ $\phi\in\mathcal{D}(I)$, 1.16.11 $\displaystyle\left\langle\delta_{x_{0}},\phi\right\rangle$ $\displaystyle=\phi(x_{0}),$ $\phi\in\mathcal{D}(I)$, 1.16.12 $\displaystyle\left\langle{\delta_{x_{0}}}^{(n)},\phi\right\rangle$ $\displaystyle=(-1)^{n}\phi^{(n)}(x_{0}),$ $\phi\in\mathcal{D}(I)$.

The Dirac delta distribution is singular.

## §1.16(iv) Heaviside Function

 1.16.13 $\displaystyle H\left(x\right)$ $\displaystyle=\begin{cases}1,&x>0,\\ 0,&x\leq 0.\end{cases}$ ⓘ Defines: $H\left(\NVar{x}\right)$: Heaviside function Referenced by: §1.14(iii), §1.16(viii) Permalink: http://dlmf.nist.gov/1.16.E13 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1 1.16.14 $\displaystyle H\left(x-x_{0}\right)$ $\displaystyle=\begin{cases}1,&x>x_{0},\\ 0,&x\leq x_{0}.\end{cases}$ ⓘ Symbols: $H\left(\NVar{x}\right)$: Heaviside function Permalink: http://dlmf.nist.gov/1.16.E14 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1 1.16.15 $\displaystyle D\!H$ $\displaystyle=\delta,$ ⓘ Symbols: $H\left(\NVar{x}\right)$: Heaviside function, $\delta_{x}$: Dirac delta distribution and $Df$: distributional derivative Referenced by: §1.16(viii) Permalink: http://dlmf.nist.gov/1.16.E15 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1 1.16.16 $\displaystyle D\!H\left(x-x_{0}\right)$ $\displaystyle=\delta_{x_{0}}.$ ⓘ Symbols: $H\left(\NVar{x}\right)$: Heaviside function, $\delta_{x}$: Dirac delta distribution and $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E16 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1

Suppose $f(x)$ is infinitely differentiable except at $x_{0}$, where left and right derivatives of all orders exist, and

 1.16.17 $\sigma_{n}=f^{(n)}(x_{0}+)-f^{(n)}(x_{0}-).$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.16.E17 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1

Then

 1.16.18 $D^{m}f=f^{(m)}+\sigma_{0}{\delta_{x_{0}}}^{(m-1)}+\sigma_{1}{\delta_{x_{0}}}^{% (m-2)}+\dots+\sigma_{m-1}\delta_{x_{0}},$ $m=1,2,\dots$. ⓘ Symbols: $\delta_{x}$: Dirac delta distribution, $m$: nonnegative integer and $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E18 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1

For $\alpha>-1$,

 1.16.19 $x^{\alpha}_{+}=x^{\alpha}H\left(x\right)=\begin{cases}x^{\alpha},&x>0,\\ 0,&x\leq 0.\end{cases}$ ⓘ Symbols: $H\left(\NVar{x}\right)$: Heaviside function Permalink: http://dlmf.nist.gov/1.16.E19 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1

For $\alpha>0$,

 1.16.20 $Dx^{\alpha}_{+}=\alpha x_{+}^{\alpha-1}.$ ⓘ Symbols: $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E20 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1

For $\alpha<-1$ and $\alpha$ not an integer, define

 1.16.21 $x^{\alpha}_{+}=\frac{1}{{\left(\alpha+1\right)_{n}}}D^{n}x_{+}^{\alpha+n},$ ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E21 Encodings: TeX, pMML, png Correction (effective with 1.1.2): The Pochhammer symbol now links to its definition. See also: Annotations for §1.16(iv), §1.16 and Ch.1

where $n$ is an integer such that $\alpha+n>-1$. Similarly, we write

 1.16.22 $\ln_{+}x=H\left(x\right)\ln x=\begin{cases}\ln x,&x>0,\\ 0,&x\leq 0,\end{cases}$ ⓘ Symbols: $H\left(\NVar{x}\right)$: Heaviside function and $\ln\NVar{z}$: principal branch of logarithm function Permalink: http://dlmf.nist.gov/1.16.E22 Encodings: TeX, pMML, png See also: Annotations for §1.16(iv), §1.16 and Ch.1

and define

 1.16.23 $(-1)^{n}n!x_{+}^{-1-n}=D^{(n+1)}\ln_{+}x,$ $n=0,1,2,\dots$.

## §1.16(v) Tempered Distributions

The space $\mathcal{T}(\mathbb{R})$ of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are $O\left(|x|^{-N}\right)$ as $|x|\to\infty$ for all $N$.

A sequence $\{\phi_{n}\}$ of functions in $\mathcal{T}$ is said to converge to a function $\phi\in\mathcal{T}$ as $n\to\infty$ if the sequence $\{\phi_{n}^{(k)}\}$ converges uniformly to $\phi^{(k)}$ on every finite interval and if the constants $c_{k,N}$ in the inequalities

 1.16.24 $|x^{N}\phi_{n}^{(k)}|\leq c_{k,N}$ ⓘ Symbols: $k$: integer, $n$: nonnegative integer and $\phi(x)$: test function Permalink: http://dlmf.nist.gov/1.16.E24 Encodings: TeX, pMML, png See also: Annotations for §1.16(v), §1.16 and Ch.1

do not depend on $n$.

A tempered distribution is a continuous linear functional $\Lambda$ on $\mathcal{T}$. (See the definition of a distribution in §1.16(i).) The set of tempered distributions is denoted by $\mathcal{T}^{*}$.

A sequence of tempered distributions $\Lambda_{n}$ converges to $\Lambda$ in $\mathcal{T}^{*}$ if

 1.16.25 $\lim_{n\to\infty}\left\langle\Lambda_{n},\phi\right\rangle=\left\langle\Lambda% ,\phi\right\rangle,$

for all $\phi\in\mathcal{T}$.

The derivatives of tempered distributions are defined in the same way as derivatives of distributions.

For a detailed discussion of tempered distributions see Lighthill (1958).

## §1.16(vi) Distributions of Several Variables

Let $\mathcal{D}({\mathbb{R}}^{n})=\mathcal{D}_{n}$ be the set of all infinitely differentiable functions in $n$ variables, $\phi(x_{1},x_{2},\dots,x_{n})$, with compact support in ${\mathbb{R}}^{n}$. If $k=(k_{1},\dots,k_{n})$ is a multi-index and $x=(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}$, then we write $x^{k}=x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}$ and $\phi^{(k)}(x)={\partial}^{k}\phi/(\partial x_{1}^{k_{1}}\cdots\partial x_{n}^{% k_{n}})$. A sequence $\{\phi_{m}\}$ of functions in $\mathcal{D}_{n}$ converges to a function $\phi\in\mathcal{D}_{n}$ if the supports of $\phi_{m}$ lie in a fixed compact subset $K$ of ${\mathbb{R}}^{n}$ and $\phi_{m}^{(k)}$ converges uniformly to $\phi^{(k)}$ in $K$ for every multi-index $k=(k_{1},k_{2},\dots,k_{n})$. A distribution in ${\mathbb{R}}^{n}$ is a continuous linear functional on $\mathcal{D}_{n}$.

The partial derivatives of distributions in ${\mathbb{R}}^{n}$ can be defined as in §1.16(ii). A locally integrable function $f(x)=f(x_{1},x_{2},\dots,x_{n})$ gives rise to a distribution $\Lambda_{f}$ defined by

 1.16.26 $\left\langle\Lambda_{f},\phi\right\rangle=\int_{{\mathbb{R}}^{n}}f(x)\phi(x)% \mathrm{d}x,$ $\phi\in\mathcal{D}_{n}$.

The distributional derivative $D^{k}f$ of $f$ is defined by

 1.16.27 $\left\langle D^{k}f,\phi\right\rangle=(-1)^{\left|k\right|}\int_{{\mathbb{R}}^% {n}}f(x)\phi^{(k)}(x)\mathrm{d}x,$ $\phi\in\mathcal{D}_{n}$,

where $k$ is a multi-index and $\left|k\right|=k_{1}+k_{2}+\dots+k_{n}$.

For tempered distributions the space of test functions $\mathcal{T}_{n}$ is the set of all infinitely-differentiable functions $\phi$ of $n$ variables that satisfy

 1.16.28 $|x^{m}\phi^{(k)}(x)|\leq c_{m,k},$ $x\in{\mathbb{R}}^{n}$.

Here $m=(m_{1},m_{2},\dots,m_{n})$ and $k=(k_{1},k_{2},\dots,k_{n})$ are multi-indices, and $c_{m,k}$ are constants. Tempered distributions are continuous linear functionals on this space of test functions. The space of tempered distributions is denoted by $\mathcal{T}^{*}_{n}$.

## §1.16(vii) Fourier Transforms of Tempered Distributions

Suppose $\phi$ is a test function in $\mathcal{T}_{n}$. Then its Fourier transform is

1.16.29 $\mathscr{F}(\phi)(\mathbf{x})=\mathscr{F}\phi(\mathbf{x})=\frac{1}{(2\pi)^{n/2% }}\int_{{\mathbb{R}}^{n}}\phi(\mathbf{t})e^{i\mathbf{x}\cdot\mathbf{t}}\mathrm% {d}\mathbf{t},$

where $\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$ and $\mathbf{x}\cdot\mathbf{t}=x_{1}t_{1}+\dots+x_{n}t_{n}$. $\mathscr{F}\phi(\mathbf{x})$ is also in $\mathcal{T}_{n}$.

Let

1.16.30 $\mathbf{D}=\left(\frac{1}{\mathrm{i}}\frac{\partial}{\partial x_{1}},\frac{1}{% \mathrm{i}}\frac{\partial}{\partial x_{2}},\ldots,\frac{1}{\mathrm{i}}\frac{% \partial}{\partial x_{n}}\right).$

For a multi-index $\boldsymbol{{\alpha}}=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$, define

 1.16.31 $P(\mathbf{x})=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\mathbf{x}^% {\boldsymbol{{\alpha}}}=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}x% _{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}},$ ⓘ Symbols: $n$: nonnegative integer and $P$: polynomial of several variables Permalink: http://dlmf.nist.gov/1.16.E31 Encodings: TeX, pMML, png See also: Annotations for §1.16(vii), §1.16 and Ch.1

and

1.16.32 $P(\mathbf{D})=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\mathbf{D}^% {\alpha}=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\left(\frac{1}{% \mathrm{i}}\frac{\partial}{\partial x_{1}}\right)^{\alpha_{1}}\dots\left(\frac% {1}{\mathrm{i}}\frac{\partial}{\partial x_{n}}\right)^{\alpha_{n}}.$

Here $\boldsymbol{{\alpha}}$ ranges over a finite set of multi-indices, $P(\mathbf{x})$ is a multivariate polynomial, and $P(\mathbf{D})$ is a partial differential operator. Then

1.16.33 $\mathscr{F}(P(\mathbf{D})\phi)(\mathbf{x})=P(-\mathbf{x})\mathscr{F}\phi(% \mathbf{x}),$

and

1.16.34 $\mathscr{F}(P\phi)(\mathbf{x})=P(\mathbf{D})\mathscr{F}\phi(\mathbf{x}).$

If $u\in\mathcal{T}^{*}_{n}$ is a tempered distribution, then its Fourier transform $\mathscr{F}\left(u\right)$ is defined by

1.16.35 $\left\langle\mathscr{F}\left(u\right),\phi\right\rangle=\left\langle u,% \mathscr{F}(\phi)\right\rangle,$
$\phi\in\mathcal{T}_{n}$.

The Fourier transform $\mathscr{F}\left(u\right)$ of a tempered distribution is again a tempered distribution, and

1.16.36 $\left\langle\mathscr{F}\left(P(\mathbf{D})u\right),\phi\right\rangle=\left% \langle P_{-}\mathscr{F}\left(u\right),\phi\right\rangle=\left\langle\mathscr{% F}\left(u\right),P_{-}\phi\right\rangle,$
1.16.37 $\left\langle\mathscr{F}\left(Pu\right),\phi\right\rangle=\left\langle P(% \mathbf{D})\mathscr{F}\left(u\right),\phi\right\rangle,$

in which $P_{-}(\mathbf{x})=P(-\mathbf{x})$; compare (1.16.33) and (1.16.34). In (1.16.36) and (1.16.37) the derivatives in $P(\mathbf{D})$ are understood to be in the sense of distributions, as defined in §1.16(ii) and we also use the convention (1.16.6).

## §1.16(viii) Fourier Transforms of Special Distributions

We use the notation of the previous subsection and take $n=1$ and $u=\delta$ in (1.16.35). We obtain

 1.16.38 $\left\langle\mathscr{F}\left(\delta\right),\phi\right\rangle=\left\langle% \delta,\mathscr{F}(\phi)\right\rangle=\left\langle\delta,\frac{1}{\sqrt{2\pi}}% \int^{\infty}_{-\infty}\phi(t){\mathrm{e}}^{\mathrm{i}xt}\mathrm{d}t\right% \rangle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi(t)\mathrm{d}t=\frac{1% }{\sqrt{2\pi}}\left\langle 1,\phi\right\rangle,$ $\phi\in\mathcal{T}$.

As distributions, the last equation reads

 1.16.39 $\mathscr{F}\left(\delta\right)=\frac{1}{\sqrt{2\pi}},$

which is often written conventionally as

 1.16.40 $\int^{\infty}_{-\infty}\delta\left(t\right){\mathrm{e}}^{\mathrm{i}xt}\mathrm{% d}t=1;$

Since $\sqrt{2\pi}\mathscr{F}\left(\delta\right)=1$, we have

 1.16.41 $\left\langle\mathscr{F}\left(1\right),\phi\right\rangle=\sqrt{2\pi}\left% \langle\mathscr{F}\left(\mathscr{F}\left(\delta\right)\right),\phi\right% \rangle=\sqrt{2\pi}\left\langle\mathscr{F}\left(\delta\right),\mathscr{F}(\phi% )\right\rangle=\sqrt{2\pi}\left\langle\delta,\mathscr{F}(\mathscr{F}(\phi))% \right\rangle=\sqrt{2\pi}\left\langle\delta,\phi_{-}\right\rangle=\sqrt{2\pi}% \phi(0),$

in which $\phi_{-}(x)=\phi(-x)$. The second to last equality follows from the Fourier integral formula (1.17.8). Since the quantity on the extreme right of (1.16.41) is equal to $\sqrt{2\pi}\left\langle\delta,\phi\right\rangle$, as distributions, the result in this equation can be stated as

 1.16.42 $\mathscr{F}\left(1\right)=\sqrt{2\pi}\delta,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\delta_{x}$: Dirac delta distribution and $\mathscr{F}\left(\NVar{u}\right)$: Fourier transform of a tempered distribution Referenced by: §1.16(viii) Permalink: http://dlmf.nist.gov/1.16.E42 Encodings: TeX, pMML, png See also: Annotations for §1.16(viii), §1.16 and Ch.1

and conventionally it is expressed as

 1.16.43 $\frac{1}{2\pi}\int^{\infty}_{-\infty}{\mathrm{e}}^{\mathrm{i}xt}\mathrm{d}t=% \delta\left(x\right);$

It is easily verified that

 1.16.44 $\operatorname{sign}\left(x\right)=2H\left(x\right)-1,$ $x\neq 0$, ⓘ Symbols: $H\left(\NVar{x}\right)$: Heaviside function and $\operatorname{sign}\NVar{x}$: sign of Permalink: http://dlmf.nist.gov/1.16.E44 Encodings: TeX, pMML, png See also: Annotations for §1.16(viii), §1.16 and Ch.1

and from (1.16.15) we find

 1.16.45 ${\operatorname{sign}}^{\prime}=2H'=2\delta,$ ⓘ Symbols: $H\left(\NVar{x}\right)$: Heaviside function, $\delta_{x}$: Dirac delta distribution and $\operatorname{sign}\NVar{x}$: sign of Permalink: http://dlmf.nist.gov/1.16.E45 Encodings: TeX, pMML, png See also: Annotations for §1.16(viii), §1.16 and Ch.1

where $H\left(x\right)$ is the Heaviside function defined in (1.16.13), and the derivatives are to be understood in the sense of distributions. Then

 1.16.46 $\mathscr{F}\left({\operatorname{sign}}^{\prime}\right)=\mathscr{F}\left(2H'% \right)=2\mathscr{F}\left(\delta\right)=\sqrt{\frac{2}{\pi}},$

and from (1.16.36) with $u=\operatorname{sign}$, $P(\mathbf{D})=D$, and $P_{-}(x)=-ix$, we have also

 1.16.47 $\mathscr{F}\left({\operatorname{sign}}^{\prime}\right)=\frac{x}{\mathrm{i}}% \mathscr{F}\left(\operatorname{sign}\right).$ ⓘ Symbols: $\mathscr{F}\left(\NVar{u}\right)$: Fourier transform of a tempered distribution, $\mathrm{i}$: imaginary unit and $\operatorname{sign}\NVar{x}$: sign of Referenced by: §1.16(viii) Permalink: http://dlmf.nist.gov/1.16.E47 Encodings: TeX, pMML, png See also: Annotations for §1.16(viii), §1.16 and Ch.1

Coupling (1.16.46) and (1.16.47) gives

 1.16.48 $\mathscr{F}\left(\operatorname{sign}\right)=\sqrt{\frac{2}{\pi}}\,\frac{% \mathrm{i}}{x},$

that is

 1.16.49 $\left\langle\mathscr{F}\left(\operatorname{sign}\right),\phi\right\rangle=% \mathrm{i}\sqrt{\frac{2}{\pi}}\pvint^{\infty}_{-\infty}\frac{\phi(x)}{x}% \mathrm{d}x.$

The Fourier transform of $H\left(x\right)$ now follows from (1.16.42) and (1.16.48). Indeed, we have

 1.16.50 $\mathscr{F}\left(H\right)=\frac{1}{2}\mathscr{F}\left(1+\operatorname{sign}% \right)=\frac{1}{2}\left[\mathscr{F}\left(1\right)+\mathscr{F}\left(% \operatorname{sign}\right)\right]=\sqrt{\frac{\pi}{2}}\left(\delta+\frac{% \mathrm{i}}{\pi x}\right),$

that is

 1.16.51 $\left\langle\mathscr{F}\left(H\right),\phi\right\rangle=\sqrt{\frac{\pi}{2}}% \phi(0)+\frac{\mathrm{i}}{\sqrt{2\pi}}\pvint^{\infty}_{-\infty}\frac{\phi(x)}{% x}\mathrm{d}x.$

For more detailed discussions of the formulas in this section, see Kanwal (1983) and Debnath and Bhatta (2015).