Editing Dirac delta function (section)

==Derivatives==
The derivative of the Dirac delta distribution, denoted {{math|''δ&prime;''}} and also called the ''Dirac delta prime'' or ''Dirac delta derivative'' as described in [[Laplacian of the indicator]], is defined on compactly supported smooth test functions {{mvar|φ}} by{{sfn|Gelfand|Shilov|1966–1968|p=26}}
<math display="block">\delta'[\varphi] = -\delta[\varphi']=-\varphi'(0).</math>

The first equality here is a kind of [[integration by parts]], for if {{mvar|δ}} were a true function then
<math display="block">\int_{-\infty}^\infty \delta'(x)\varphi(x)\,dx
= \delta(x)\varphi(x)|_{-\infty}^{\infty} -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx
= -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx
= -\varphi'(0).</math>

By [[mathematical induction]], the {{mvar|k}}-th derivative of {{mvar|δ}} is defined similarly as the distribution given on test functions by

<math display="block">\delta^{(k)}[\varphi] = (-1)^k \varphi^{(k)}(0).</math>

In particular, {{mvar|δ}} is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:{{sfn|Gelfand|Shilov|1966–1968|loc=§2.1}}
<math display="block">\delta'(x) = \lim_{h\to 0} \frac{\delta(x+h)-\delta(x)}{h}.</math>

More properly, one has
<math display="block">\delta' = \lim_{h\to 0} \frac{1}{h}(\tau_h\delta - \delta)</math>
where {{mvar|τ<sub>h</sub>}} is the translation operator, defined on functions by {{math|1=''τ<sub>h</sub>φ''(''x'') = ''φ''(''x'' + ''h'')}}, and on a distribution {{mvar|S}} by
<math display="block">(\tau_h S)[\varphi] = S[\tau_{-h}\varphi].</math>

In the theory of [[electromagnetism]], the first derivative of the delta function represents a point magnetic [[dipole]] situated at the origin.  Accordingly, it is referred to as a dipole or the [[unit doublet|doublet function]].<ref>{{MathWorld|title=Doublet Function|urlname=DoubletFunction}}</ref>

The derivative of the delta function satisfies a number of basic properties, including:{{sfn|Bracewell|2000|p=86}}
<math display="block">
\begin{align}
 \delta'(-x) &= -\delta'(x) \\
 x\delta'(x) &= -\delta(x)
\end{align}
</math>
which can be shown by applying a test function and integrating by parts.

The latter of these properties can also be demonstrated by applying distributional derivative definition, Leibniz&nbsp;'s theorem and linearity of inner product:<ref>{{Cite web|url=https://www.matematicamente.it/forum/viewtopic.php?f=36&t=62388&start=10#wrap|title=Gugo82's comment on the distributional derivative of Dirac's delta|date=12 September 2010|website=matematicamente.it}}</ref>

<math display="block">
\begin{align}
\langle x\delta', \varphi \rangle \, &= \, \langle \delta', x\varphi \rangle \, = \, -\langle\delta,(x\varphi)'\rangle \, = \, - \langle \delta, x'\varphi + x\varphi'\rangle  \, = \, - \langle \delta, x'\varphi\rangle  - \langle\delta, x\varphi'\rangle  \, =  \,
- x'(0)\varphi(0) - x(0)\varphi'(0) \\
&= \, -x'(0) \langle \delta , \varphi \rangle - x(0) \langle \delta, \varphi' \rangle \, = \, -x'(0) \langle \delta,\varphi\rangle + x(0) \langle \delta', \varphi \rangle \, = \,
\langle x(0)\delta' - x'(0)\delta, \varphi \rangle \\
 \Longrightarrow x(t)\delta'(t) &= x(0)\delta'(t) - x'(0)\delta(t) = -x'(0)\delta(t) = -\delta(t)
\end{align}
</math>

Furthermore, the convolution of {{mvar|δ&prime;}} with a compactly-supported, smooth function {{mvar|f}} is

<math display="block">\delta'*f = \delta*f' = f',</math>

which follows from the properties of the distributional derivative of a convolution.

===Higher dimensions===
More generally, on an [[open set]] {{mvar|U}} in the {{mvar|n}}-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math>, the Dirac delta distribution centered at a point {{math|''a'' ∈ ''U''}} is defined by{{sfn|Hörmander|1983|p=56}}
<math display="block">\delta_a[\varphi]=\varphi(a)</math>
for all <math>\varphi \in C_c^\infty(U)</math>, the space of all smooth functions with compact support on {{mvar|U}}.  If <math>\alpha = (\alpha_1, \ldots, \alpha_n)</math> is any [[multi-index]] with <math> |\alpha|=\alpha_1+\cdots+\alpha_n</math> and <math>\partial^\alpha</math> denotes the associated mixed [[partial derivative]] operator, then the {{mvar|α}}-th derivative {{mvar|∂<sup>α</sup>δ<sub>a</sub>}} of {{mvar|δ<sub>a</sub>}} is given by{{sfn|Hörmander|1983|p=56}}

<math display="block">\left\langle \partial^\alpha \delta_{a}, \, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = (-1)^{| \alpha |} \partial^\alpha \varphi (x) \Big|_{x = a} \quad \text{ for all } \varphi \in C_c^\infty(U).</math>

That is, the {{mvar|α}}-th derivative of {{mvar|δ<sub>a</sub>}} is the distribution whose value on any test function {{mvar|φ}} is the {{mvar|α}}-th derivative of {{mvar|φ}} at {{mvar|a}} (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as [[double layer potential|double layers]] along the coordinate planes.  More generally, the [[normal derivative]] of a simple layer supported on a surface is a double layer supported on that surface and represents a laminar magnetic monopole.  Higher derivatives of the delta function are known in physics as [[multipole]]s.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support.  If {{mvar|S}} is any distribution on {{mvar|U}} supported on the set {{math|{{brace|''a''}}}} consisting of a single point, then there is an integer {{mvar|m}} and coefficients {{mvar|c<sub>α</sub>}} such that{{sfn|Hörmander|1983|p=56}}{{sfn|Rudin|1991|loc=Theorem 6.25}}
<math display="block">S = \sum_{|\alpha|\le m} c_\alpha \partial^\alpha\delta_a.</math>