§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 $x^{\alpha}_{+}=\frac{1}{(\alpha+1)_{n}}D^{n}x_{+}^{\alpha+n},$ ⓘ Symbols: $n$: nonnegative integer and $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E21 Encodings: TeX, pMML, png 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 $x$ 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 $x$ 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 and $\operatorname{sign}\NVar{x}$: sign of $x$ 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).