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Dirac delta function

In mathematical analysis, the Dirac delta function (or $δ$ distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.

The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.

Motivation and overview
The graph of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis. The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar abstractions such as a point charge, point mass or electron point. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).

To be specific, suppose that a billiard ball is at rest. At time $$t=0$$ it is struck by another ball, imparting it with a momentum $P$, with units kg⋅m⋅s−1. The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The force therefore is $P δ(t)$; the units of $δ(t)$ are s−1.

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval $\Delta t = [0,T]$. That is,

$$F_{\Delta t}(t) = \begin{cases} P/\Delta t& 0 0\\ 0 & t < 0.\end{cases}$$

Here the functions $$F_{\Delta t}$$ are thought of as useful approximations to the idea of instantaneous transfer of momentum.

The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of pointwise convergence) $\lim_{\Delta t\to 0^+}F_{\Delta t}$ is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property

$$\int_{-\infty}^\infty F_{\Delta t}(t)\,dt = P,$$

which holds for all $\Delta t>0$, should continue to hold in the limit. So, in the equation $F(t)=P\,\delta(t)=\lim_{\Delta t\to 0}F_{\Delta t}(t)$, it is understood that the limit is always taken.

In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers. For example, the objects $f(x) = δ(x)$ and $g(x) = 0$ are equal everywhere except at $x = 0$ yet have integrals that are different. According to Lebesgue integration theory, if $f$ and $g$ are functions such that $f = g$ almost everywhere, then $f$ is integrable if and only if $g$ is integrable and the integrals of $f$ and $g$ are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions.

History
Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:

$$f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty\ \ d\alpha \, f(\alpha) \ \int_{-\infty}^\infty dp\ \cos (px-p\alpha)\, $$

which is tantamount to the introduction of the $δ$-function in the form:

$$\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty dp\ \cos (px-p\alpha) \. $$

Later, Augustin Cauchy expressed the theorem using exponentials:

$$f(x)=\frac{1}{2\pi} \int_{-\infty} ^ \infty \ e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp. $$

Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).

As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as

$$\begin{align} f(x)&=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp \\[4pt] &=\frac{1}{2\pi} \int_{-\infty}^\infty \left(\int_{-\infty}^\infty e^{ipx} e^{-ip\alpha } \,dp \right)f(\alpha)\,d \alpha =\int_{-\infty}^\infty \delta (x-\alpha) f(\alpha) \,d \alpha, \end{align}$$

where the δ-function is expressed as

$$\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\,dp \. $$

A rigorous interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:
 * The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.

Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L2-theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945) ...", and leading to the formal development of the Dirac delta function.

An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin-Louis Cauchy. Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of the Quantum Dynamics and used in his textbook The Principles of Quantum Mechanics. He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.

Definitions
The Dirac delta function $$\delta (x)$$ can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

$$\delta(x) \simeq \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$$

and which is also constrained to satisfy the identity

$$\int_{-\infty}^\infty \delta(x) \, dx = 1.$$

This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no extended real number valued function defined on the real numbers has these properties.

As a measure
One way to rigorously capture the notion of the Dirac delta function is to define a measure, called Dirac measure, which accepts a subset $A$ of the real line $R$ as an argument, and returns $δ(A) = 1$ if $0 ∈ A$, and $δ(A) = 0$ otherwise. If the delta function is conceptualized as modeling an idealized point mass at 0, then $δ(A)$ represents the mass contained in the set $A$. One may then define the integral against $δ$ as the integral of a function against this mass distribution. Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure $δ$ satisfies

$$\int_{-\infty}^\infty f(x) \, \delta(dx) = f(0)$$

for all continuous compactly supported functions $f$. The measure $δ$ is not absolutely continuous with respect to the Lebesgue measure—in fact, it is a singular measure. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property

$$\int_{-\infty}^\infty f(x)\, \delta(x)\, dx = f(0)$$

holds. As a result, the latter notation is a convenient abuse of notation, and not a standard (Riemann or Lebesgue) integral.

As a probability measure on $R$, the delta measure is characterized by its cumulative distribution function, which is the unit step function.

$$H(x) = \begin{cases} 1 & \text{if } x\ge 0\\ 0 & \text{if } x < 0. \end{cases}$$

This means that $H(x)$ is the integral of the cumulative indicator function $1_{(−∞, x]}$ with respect to the measure $δ$; to wit,

$$H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta(dt) = \delta\!\left((-\infty,x]\right),$$

the latter being the measure of this interval. Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:

$$\int_{-\infty}^\infty f(x)\,\delta(dx) = \int_{-\infty}^\infty f(x) \,dH(x).$$

All higher moments of $δ$ are zero. In particular, characteristic function and moment generating function are both equal to one.

As a distribution
In the theory of distributions, a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them. In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" test function $φ$. Test functions are also known as bump functions. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.

A typical space of test functions consists of all smooth functions on $R$ with compact support that have as many derivatives as required. As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by

for every test function $$.

For $φ$ to be properly a distribution, it must be continuous in a suitable topology on the space of test functions. In general, for a linear functional $δ$ on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer $S$ there is an integer $M_{N}$ and a constant $N$ such that for every test function $C_{N}$, one has the inequality

$$\left|S[\varphi]\right| \le C_N \sum_{k=0}^{M_N}\sup_{x\in [-N,N]} \left|\varphi^{(k)}(x)\right|$$

where $sup$ represents the supremum. With the $φ$ distribution, one has such an inequality (with $C_{N} = 1)$ with $M_{N} = 0$ for all $δ$. Thus $N$ is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being $\{0\}$).

The delta distribution can also be defined in several equivalent ways. For instance, it is the distributional derivative of the Heaviside step function. This means that for every test function $δ$, one has

$$\delta[\varphi] = -\int_{-\infty}^\infty \varphi'(x)\,H(x)\,dx.$$

Intuitively, if integration by parts were permitted, then the latter integral should simplify to

$$\int_{-\infty}^\infty \varphi(x)\,H'(x)\,dx = \int_{-\infty}^\infty \varphi(x)\,\delta(x)\,dx,$$

and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have

$$-\int_{-\infty}^\infty \varphi'(x)\,H(x)\, dx = \int_{-\infty}^\infty \varphi(x)\,dH(x).$$

In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation ($φ$) defines a Daniell integral on the space of all compactly supported continuous functions $$ which, by the Riesz representation theorem, can be represented as the Lebesgue integral of $φ$ with respect to some Radon measure.

Generally, when the term Dirac delta function is used, it is in the sense of distributions rather than measures, the Dirac measure being among several terms for the corresponding notion in measure theory. Some sources may also use the term Dirac delta distribution.

Generalizations
The delta function can be defined in $φ$-dimensional Euclidean space $R^{n}$ as the measure such that

$$\int_{\mathbf{R}^n} f(\mathbf{x})\,\delta(d\mathbf{x}) = f(\mathbf{0})$$

for every compactly supported continuous function $n$. As a measure, the $f$-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with $x = (x_{1}, x_{2}, ..., x_{n})$, one has

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case. However, despite widespread use in engineering contexts, ($n$) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.

The notion of a Dirac measure makes sense on any set. Thus if $$ is a set, $x_{0} ∈ X$ is a marked point, and $Σ$ is any sigma algebra of subsets of $$, then the measure defined on sets $A ∈ Σ$ by

$$\delta_{x_0}(A)=\begin{cases} 1 &\text{if }x_0\in A\\ 0 &\text{if }x_0\notin A \end{cases}$$

is the delta measure or unit mass concentrated at $x_{0}$.

Another common generalization of the delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the differentiable structure. The delta function on a manifold $X$ centered at the point $x_{0} ∈ M$ is defined as the following distribution:

for all compactly supported smooth real-valued functions $X$ on $M$. A common special case of this construction is a case in which $$ is an open set in the Euclidean space $R^{n}$.

On a locally compact Hausdorff space $φ$, the Dirac delta measure concentrated at a point $M$ is the Radon measure associated with the Daniell integral ($M$) on compactly supported continuous functions $X$. At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping $$x_0\mapsto \delta_{x_0}$$ is a continuous embedding of $x$ into the space of finite Radon measures on $$, equipped with its vague topology. Moreover, the convex hull of the image of $φ$ under this embedding is dense in the space of probability measures on $X$.

Scaling and symmetry
The delta function satisfies the following scaling property for a non-zero scalar $X$:

$$\int_{-\infty}^\infty \delta(\alpha x)\,dx =\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|} =\frac{1}{|\alpha|}$$

and so

Scaling property proof: $$\int\limits_{-\infty}^{\infty} dx\ g(x) \delta (ax) = \frac{1}{a}\int\limits_{-\infty}^{\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') $$ where a change of variable $x′ = ax$ is used. If $X$ is negative, i.e., $a = −|a|$, then $$\int\limits_{-\infty}^{\infty} dx\ g(x) \delta (ax) = \frac{1}{-\left \vert a \right \vert}\int\limits_{\infty}^{-\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') = \frac{1}{\left \vert a \right \vert}\int\limits_{-\infty}^{\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') = \frac{1}{\left \vert a \right \vert}g(0). $$ Thus, $\delta (ax) = \frac{1}{\left \vert a \right \vert} \delta(x)$.

In particular, the delta function is an even distribution (symmetry), in the sense that

$$\delta(-x) = \delta(x)$$

which is homogeneous of degree $−1$.

Algebraic properties
The distributional product of $X$ with $α$ is equal to zero:

$$x\,\delta(x) = 0.$$

More generally, $$(x-a)^n\delta(x-a) =0$$ for all positive integers $$n$$.

Conversely, if $xf(x) = xg(x)$, where $$ and $a$ are distributions, then

$$f(x) = g(x) +c \delta(x)$$

for some constant $δ$.

Translation
The integral of the time-delayed Dirac delta is

$$\int_{-\infty}^\infty f(t) \,\delta(t-T)\,dt = f(T).$$

This is sometimes referred to as the sifting property or the sampling property. The delta function is said to "sift out" the value of f(t) at t = T.

It follows that the effect of convolving a function $f(t)$ with the time-delayed Dirac delta $$ \delta_T(t) = \delta(t-T)$$ is to time-delay $f(t)$ by the same amount:

$$\begin{align} (f * \delta_T)(t) \ &\stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty f(\tau)\, \delta(t-T-\tau) \, d\tau \\ &= \int_{-\infty}^\infty f(\tau) \,\delta(\tau-(t-T)) \,d\tau \qquad \text{since}~ \delta(-x) = \delta(x)  \text{by (4)}\\ &= f(t-T). \end{align}$$

The sifting property holds under the precise condition that $x$ be a tempered distribution (see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution sense)

$$\int_{-\infty}^\infty \delta (\xi-x) \delta(x-\eta) \,dx = \delta(\eta-\xi).$$

Composition with a function
More generally, the delta distribution may be composed with a smooth function $g(x)$ in such a way that the familiar change of variables formula holds, that

$$\int_{\R} \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) \left|g'(x)\right| dx = \int_{g(\R)} \delta(u)\,f(u)\,du$$

provided that $f$ is a continuously differentiable function with $g′$ nowhere zero. That is, there is a unique way to assign meaning to the distribution $$\delta\circ g$$ so that this identity holds for all compactly supported test functions $g$. Therefore, the domain must be broken up to exclude the $g′ = 0$ point. This distribution satisfies $δ(g(x)) = 0$ if $c$ is nowhere zero, and otherwise if $f$ has a real root at $x_{0}$, then

$$\delta(g(x)) = \frac{\delta(x-x_0)}{|g'(x_0)|}.$$

It is natural therefore to the composition $δ(g(x))$ for continuously differentiable functions $g$ by

$$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$

where the sum extends over all roots (i.e., all the different ones) of $f$, which are assumed to be simple. Thus, for example

$$\delta\left(x^2-\alpha^2\right) = \frac{1}{2|\alpha|} \Big[\delta\left(x+\alpha\right)+\delta\left(x-\alpha\right)\Big].$$

In the integral form, the generalized scaling property may be written as

$$ \int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}. $$

Indefinite integral
For a constant $$a \isin \mathbb{R}$$ and a "well-behaved" arbitrary real-valued function $y(x)$, $$\displaystyle{\int}y(x)\delta(x-a)dx = y(a)H(x-a) + c,$$ where $H(x)$ is the Heaviside step function and $c$ is an integration constant.

Properties in n dimensions
The delta distribution in an $g$-dimensional space satisfies the following scaling property instead, $$\delta(\alpha\boldsymbol{x}) = |\alpha|^{-n}\delta(\boldsymbol{x}) ~,$$ so that $g$ is a homogeneous distribution of degree $−n$.

Under any reflection or rotation $g$, the delta function is invariant, $$\delta(\rho \boldsymbol{x}) = \delta(\boldsymbol{x})~.$$

As in the one-variable case, it is possible to define the composition of $g(x)$ with a bi-Lipschitz function $g: R^{n} → R^{n}$ uniquely so that the following holds $$\int_{\R^n} \delta(g(\boldsymbol{x}))\, f(g(\boldsymbol{x}))\left|\det g'(\boldsymbol{x})\right| d\boldsymbol{x} = \int_{g(\R^n)} \delta(\boldsymbol{u}) f(\boldsymbol{u})\,d\boldsymbol{u}$$ for all compactly supported functions $n$.

Using the coarea formula from geometric measure theory, one can also define the composition of the delta function with a submersion from one Euclidean space to another one of different dimension; the result is a type of current. In the special case of a continuously differentiable function $g : R^{n} → R$ such that the gradient of $δ$ is nowhere zero, the following identity holds $$\int_{\R^n} f(\boldsymbol{x}) \, \delta(g(\boldsymbol{x})) \,d\boldsymbol{x} = \int_{g^{-1}(0)}\frac{f(\boldsymbol{x})}{|\boldsymbol{\nabla}g|}\,d\sigma(\boldsymbol{x}) $$ where the integral on the right is over $g^{−1}(0)$, the $(n − 1)$-dimensional surface defined by $g(x) = 0$ with respect to the Minkowski content measure. This is known as a simple layer integral.

More generally, if $ρ$ is a smooth hypersurface of $R^{n}$, then we can associate to $δ$ the distribution that integrates any compactly supported smooth function $f$ over $g$: $$\delta_S[g] = \int_S g(\boldsymbol{s})\,d\sigma(\boldsymbol{s})$$

where $S$ is the hypersurface measure associated to $S$. This generalization is associated with the potential theory of simple layer potentials on $g$. If $S$ is a domain in $R^{n}$ with smooth boundary $σ$, then $δ_{S}$ is equal to the normal derivative of the indicator function of $S$ in the distribution sense,

$$-\int_{\R^n}g(\boldsymbol{x})\,\frac{\partial 1_D(\boldsymbol{x})}{\partial n}\,d\boldsymbol{x}=\int_S\,g(\boldsymbol{s})\, d\sigma(\boldsymbol{s}),$$

where $S$ is the outward normal. For a proof, see e.g. the article on the surface delta function.

In three dimensions, the delta function is represented in spherical coordinates by:

$$\delta(\boldsymbol{r}-\boldsymbol{r}_0) = \begin{cases} \displaystyle\frac{1}{r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)\delta(\phi-\phi_0)& x_0,y_0,z_0 \ne 0 \\ \displaystyle\frac{1}{2\pi r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)& x_0=y_0=0,\ z_0 \ne 0 \\ \displaystyle\frac{1}{4\pi r^2}\delta(r-r_0) & x_0=y_0=z_0 = 0 \end{cases}$$

Fourier transform
The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds

$$\widehat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi} \,\delta(x)dx = 1.$$

Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing $$\langle\cdot,\cdot\rangle$$ of tempered distributions with Schwartz functions. Thus $$\widehat{\delta}$$ is defined as the unique tempered distribution satisfying

$$\langle\widehat{\delta},\varphi\rangle = \langle\delta,\widehat{\varphi}\rangle$$

for all Schwartz functions $D$. And indeed it follows from this that $$\widehat{\delta}=1.$$

As a result of this identity, the convolution of the delta function with any other tempered distribution $S$ is simply $D$:

$$S*\delta = S.$$

That is to say that $n$ is an identity element for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an associative algebra with identity the delta function. This property is fundamental in signal processing, as convolution with a tempered distribution is a linear time-invariant system, and applying the linear time-invariant system measures its impulse response. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for $φ$, and once it is known, it characterizes the system completely. See.

The inverse Fourier transform of the tempered distribution $f(ξ) = 1$ is the delta function. Formally, this is expressed as $$\int_{-\infty}^\infty 1 \cdot e^{2\pi i x\xi}\,d\xi = \delta(x)$$ and more rigorously, it follows since $$\langle 1, \widehat{f}\rangle = f(0) = \langle\delta,f\rangle$$ for all Schwartz functions $S$.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on $R$. Formally, one has $$\int_{-\infty}^\infty e^{i 2\pi \xi_1 t} \left[e^{i 2\pi \xi_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (\xi_2 - \xi_1) t} \,dt = \delta(\xi_2 - \xi_1).$$

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution $$f(t) = e^{i2\pi\xi_1 t}$$ is $$\widehat{f}(\xi_2) = \delta(\xi_1-\xi_2)$$ which again follows by imposing self-adjointness of the Fourier transform.

By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to be $$ \int_{0}^{\infty}\delta(t-a)\,e^{-st} \, dt=e^{-sa}.$$

Derivatives
The derivative of the Dirac delta distribution, denoted $δ′$ 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 $S$ by $$\delta'[\varphi] = -\delta[\varphi']=-\varphi'(0).$$

The first equality here is a kind of integration by parts, for if $δ$ were a true function then $$\int_{-\infty}^\infty \delta'(x)\varphi(x)\,dx = \delta(x)\varphi(x)|_{-\infty}^{\infty} -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx = -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx = -\varphi'(0).$$

By mathematical induction, the $δ$-th derivative of $f$ is defined similarly as the distribution given on test functions by

$$\delta^{(k)}[\varphi] = (-1)^k \varphi^{(k)}(0).$$

In particular, $φ$ is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients: $$\delta'(x) = \lim_{h\to 0} \frac{\delta(x+h)-\delta(x)}{h}.$$

More properly, one has $$\delta' = \lim_{h\to 0} \frac{1}{h}(\tau_h\delta - \delta)$$ where $δ$ is the translation operator, defined on functions by $τ_{h}φ(x) = φ(x + h)$, and on a distribution $k$ by $$(\tau_h S)[\varphi] = S[\tau_{-h}\varphi].$$

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 doublet function.

The derivative of the delta function satisfies a number of basic properties, including: $$ \begin{align} \delta'(-x) &= -\delta'(x) \\ x\delta'(x) &= -\delta(x) \end{align} $$ 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, Liebniz 's theorem and linearity of inner product:

$$ \begin{align} \langle x\delta', \varphi \rangle \, &= \, \langle \delta', x\varphi \rangle \, = \, -\langle\delta,(x\varphi)'\rangle \, = \, - \langle \delta, x'\varphi + x\varphi'\rangle \, = \, - \langle \delta, x'\varphi\rangle  - \langle\delta, x\varphi'\rangle  \, =  \, - x'(0)\varphi(0) - x(0)\varphi'(0) \\ &= \, -x'(0) \langle \delta, \varphi \rangle - x(0) \langle \delta, \varphi' \rangle \, = \, -x'(0) \langle \delta,\varphi\rangle + x(0) \langle \delta', \varphi \rangle \, = \, \langle x(0)\delta' - x'(0)\delta, \varphi \rangle \\ \Longrightarrow x(t)\delta'(t) &= x(0)\delta'(t) - x'(0)\delta(t) = -x'(0)\delta(t) = -\delta(t) \end{align} $$

Furthermore, the convolution of $δ$ with a compactly-supported, smooth function $δ$ is

$$\delta'*f = \delta*f' = f',$$

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

Higher dimensions
More generally, on an open set $τ_{h}$ in the $S$-dimensional Euclidean space $$\mathbb{R}^n$$, the Dirac delta distribution centered at a point $a ∈ U$ is defined by $$\delta_a[\varphi]=\varphi(a)$$ for all $$\varphi \in C_c^\infty(U)$$, the space of all smooth functions with compact support on $δ′$. If $$\alpha = (\alpha_1, \ldots, \alpha_n)$$ is any multi-index with $$ |\alpha|=\alpha_1+\cdots+\alpha_n$$ and $$\partial^\alpha$$ denotes the associated mixed partial derivative operator, then the $f$-th derivative $U$ of $n$ is given by

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

That is, the $U$-th derivative of $α$ is the distribution whose value on any test function $∂^{α}δ_{a}$ is the $δ_{a}$-th derivative of $α$ at $δ_{a}$ (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as 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 multipoles.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If $φ$ is any distribution on $α$ supported on the set $\{a\}$ consisting of a single point, then there is an integer $φ$ and coefficients $a$ such that $$S = \sum_{|\alpha|\le m} c_\alpha \partial^\alpha\delta_a.$$

Representations of the delta function
The delta function can be viewed as the limit of a sequence of functions

$$\delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x), $$

where $η_{ε}(x)$ is sometimes called a nascent delta function. This limit is meant in a weak sense: either that

for all continuous functions $S$ having compact support, or that this limit holds for all smooth functions $U$ with compact support. The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the vague topology of measures, and the latter is convergence in the sense of distributions.

Approximations to the identity
Typically a nascent delta function $m$ can be constructed in the following manner. Let $c_{α}$ be an absolutely integrable function on $R$ of total integral $1$, and define $$\eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). $$

In $$ dimensions, one uses instead the scaling $$\eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). $$

Then a simple change of variables shows that $f$ also has integral $1$. One may show that ($f$) holds for all continuous compactly supported functions $η_{ε}$, and so $η$ converges weakly to $n$ in the sense of measures.

The $η_{ε}$ constructed in this way are known as an approximation to the identity. This terminology is because the space $L^{1}(R)$ of absolutely integrable functions is closed under the operation of convolution of functions: $f ∗ g ∈ L^{1}(R)$ whenever $$ and $f$ are in $L^{1}(R)$. However, there is no identity in $L^{1}(R)$ for the convolution product: no element $η_{ε}$ such that $f ∗ h = f$ for all $δ$. Nevertheless, the sequence $η_{ε}$ does approximate such an identity in the sense that

$$f*\eta_\varepsilon \to f \quad \text{as }\varepsilon\to 0.$$

This limit holds in the sense of mean convergence (convergence in $L^{1}$). Further conditions on the $f$, for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence almost everywhere.

If the initial $η = η_{1}$ is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing $g$ to be a suitably normalized bump function, for instance

$$\eta(x) = \begin{cases} e^{-\frac{1}{1-|x|^2}}& \text{if } |x| < 1\\ 0                    & \text{if } |x|\geq 1. \end{cases}$$

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking $η = η_{1}$ to be a hat function. With this choice of $η_{1}$, one has

$$ \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-\left|\frac{x}{\varepsilon}\right|,0 \right) $$

which are all continuous and compactly supported, although not smooth and so not a mollifier.

Probabilistic considerations
In the context of probability theory, it is natural to impose the additional condition that the initial $η_{1}$ in an approximation to the identity should be positive, as such a function then represents a probability distribution. Convolution with a probability distribution is sometimes favorable because it does not result in overshoot or undershoot, as the output is a convex combination of the input values, and thus falls between the maximum and minimum of the input function. Taking $η_{1}$ to be any probability distribution at all, and letting $η_{1}$ as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, $h$ has mean $η_{ε}(x) = η_{1}(x/ε)/ε$ and has small higher moments. For instance, if $0$ is the uniform distribution on $\left[-\frac{1}{2},\frac{1}{2}\right]$, also known as the rectangular function, then: $$ \eta_\varepsilon(x) = \frac{1}{\varepsilon}\operatorname{rect}\left(\frac{x}{\varepsilon}\right)= \begin{cases} \frac{1}{\varepsilon},&-\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2}, \\ 0,                   &\text{otherwise}. \end{cases}$$

Another example is with the Wigner semicircle distribution $$\eta_\varepsilon(x)= \begin{cases} \frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon < x < \varepsilon, \\ 0, & \text{otherwise}. \end{cases}$$

This is continuous and compactly supported, but not a mollifier because it is not smooth.

Semigroups
Nascent delta functions often arise as convolution semigroups. This amounts to the further constraint that the convolution of $f$ with $η_{ε}$ must satisfy $$\eta_\varepsilon * \eta_\delta = \eta_{\varepsilon+\delta}$$

for all $η_{1}$. Convolution semigroups in $ε, δ > 0$ that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction.

In practice, semigroups approximating the delta function arise as fundamental solutions or Green's functions to physically motivated elliptic or parabolic partial differential equations. In the context of applied mathematics, semigroups arise as the output of a linear time-invariant system. Abstractly, if A is a linear operator acting on functions of x, then a convolution semigroup arises by solving the initial value problem

$$\begin{cases} \dfrac{\partial}{\partial t}\eta(t,x) = A\eta(t,x), \quad t>0 \\[5pt] \displaystyle\lim_{t\to 0^+} \eta(t,x) = \delta(x) \end{cases}$$

in which the limit is as usual understood in the weak sense. Setting $L^{1}$ gives the associated nascent delta function.

Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.

The heat kernel
The heat kernel, defined by

$$\eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}}$$

represents the temperature in an infinite wire at time $η_{ε}(x) = η(ε, x)$, if a unit of heat energy is stored at the origin of the wire at time $t > 0$. This semigroup evolves according to the one-dimensional heat equation:

$$\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}.$$

In probability theory, $t = 0$ is a normal distribution of variance $η_{ε}$ and mean $η_{ε}(x)$. It represents the probability density at time $0$ of the position of a particle starting at the origin following a standard Brownian motion. In this context, the semigroup condition is then an expression of the Markov property of Brownian motion.

In higher-dimensional Euclidean space $t = ε$, the heat kernel is $$\eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}},$$ and has the same physical interpretation, mutatis mutandis. It also represents a nascent delta function in the sense that $R^{n}$ in the distribution sense as $η_{ε} → δ$.

The Poisson kernel
The Poisson kernel $$\eta_\varepsilon(x) = \frac{1}{\pi}\mathrm{Im}\left\{\frac{1}{x-\mathrm{i}\varepsilon}\right\}=\frac{1}{\pi} \frac{\varepsilon}{\varepsilon^2 + x^2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} \xi x-|\varepsilon \xi|}\,d\xi$$

is the fundamental solution of the Laplace equation in the upper half-plane. It represents the electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the Cauchy distribution and Epanechnikov and Gaussian kernel functions. This semigroup evolves according to the equation $$\frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x)$$

where the operator is rigorously defined as the Fourier multiplier $$\mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).$$

Oscillatory integrals
In areas of physics such as wave propagation and wave mechanics, the equations involved are hyperbolic and so may have more singular solutions. As a result, the nascent delta functions that arise as fundamental solutions of the associated Cauchy problems are generally oscillatory integrals. An example, which comes from a solution of the Euler–Tricomi equation of transonic gas dynamics, is the rescaled Airy function $$\varepsilon^{-1/3}\operatorname{Ai}\left (x\varepsilon^{-1/3} \right). $$

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the Dirichlet kernel below), rather than in the sense of measures.

Another example is the Cauchy problem for the wave equation in $ε → 0$: $$ \begin{align} c^{-2}\frac{\partial^2u}{\partial t^2} - \Delta u &= 0\\ u=0,\quad \frac{\partial u}{\partial t} = \delta &\qquad \text{for }t=0. \end{align} $$

The solution $η$ represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin.

Other approximations to the identity of this kind include the sinc function (used widely in electronics and telecommunications) $$\eta_\varepsilon(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=\frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}} \cos(kx)\,dk $$

and the Bessel function $$ \eta_\varepsilon(x) =  \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right). $$

Plane wave decomposition
One approach to the study of a linear partial differential equation $$L[u]=f,$$

where $η$ is a differential operator on $R^{1+1}$, is to seek first a fundamental solution, which is a solution of the equation $$L[u]=\delta.$$

When $η_{ε}$ is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form $$L[u]=h$$

where $η_{δ}$ is a plane wave function, meaning that it has the form $$h = h(x\cdot\xi)$$

for some vector $ε$. Such an equation can be resolved (if the coefficients of $u$ are analytic functions) by the Cauchy–Kovalevskaya theorem or (if the coefficients of $L$ are constant) by quadrature. So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations.

Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by Johann Radon, and then developed in this form by Fritz John (1955). Choose $L$ so that $R^{n}$ is an even integer, and for a real number $h$, put $$g(s) = \operatorname{Re}\left[\frac{-s^k\log(-is)}{k!(2\pi i)^n}\right] =\begin{cases} \frac{|s|^k}{4k!(2\pi i)^{n-1}}  &n \text{ odd}\\[5pt] -\frac{|s|^k\log|s|}{k!(2\pi i)^n}&n \text{ even.} \end{cases}$$

Then $ξ$ is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure $L$ of $n + k$ for $L$ in the unit sphere $g(x · ξ)$: $$\delta(x) = \Delta_x^{(n+k)/2} \int_{S^{n-1}} g(x\cdot\xi)\,d\omega_\xi.$$

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function $k$, $$\varphi(x) = \int_{\mathbf{R}^n}\varphi(y)\,dy\,\Delta_x^{\frac{n+k}{2}} \int_{S^{n-1}} g((x-y)\cdot\xi)\,d\omega_\xi.$$

The result follows from the formula for the Newtonian potential (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the Radon transform because it recovers the value of $S^{n−1}$ from its integrals over hyperplanes. For instance, if $s$ is odd and $φ(x)$, then the integral on the right hand side is $$ \begin{align} & c_n \Delta^{\frac{n+1}{2}}_x\iint_{S^{n-1}} \varphi(y)|(y-x) \cdot \xi| \, d\omega_\xi \, dy \\[5pt] & \qquad = c_n \Delta^{(n+1)/2}_x \int_{S^{n-1}} \, d\omega_\xi \int_{-\infty}^\infty |p| R\varphi(\xi,p+x\cdot\xi)\,dp \end{align} $$

where $k = 1$ is the Radon transform of $δ$: $$R\varphi(\xi,p) = \int_{x\cdot\xi=p} f(x)\,d^{n-1}x.$$

An alternative equivalent expression of the plane wave decomposition is: $$\delta(x) = \begin{cases} \frac{(n-1)!}{(2\pi i)^n}\displaystyle\int_{S^{n-1}}(x\cdot\xi)^{-n} \, d\omega_\xi & n\text{ even} \\ \frac{1}{2(2\pi i)^{n-1}}\displaystyle\int_{S^{n-1}}\delta^{(n-1)}(x\cdot\xi)\,d\omega_\xi & n\text{ odd}. \end{cases}$$

Fourier kernels
In the study of Fourier series, a major question consists of determining whether and in what sense the Fourier series associated with a periodic function converges to the function. The $dω$-th partial sum of the Fourier series of a function $ξ$ of period $Rφ(ξ, p)$ is defined by convolution (on the interval $φ$) with the Dirichlet kernel: $$D_N(x) = \sum_{n=-N}^N e^{inx} = \frac{\sin\left(\left(N+\frac12\right)x\right)}{\sin(x/2)}.$$ Thus, $$s_N(f)(x) = D_N*f(x) = \sum_{n=-N}^N a_n e^{inx}$$ where $$a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy.$$ A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval $n$ tends to a multiple of the delta function as $2π$. This is interpreted in the distribution sense, that $$s_N(f)(0) = \int_{-\pi}^{\pi} D_N(x)f(x)\,dx \to 2\pi f(0)$$ for every compactly supported function $φ$. Thus, formally one has $$\delta(x) = \frac1{2\pi} \sum_{n=-\infty}^\infty e^{inx}$$ on the interval $n$.

Despite this, the result does not hold for all compactly supported functions: that is $N → ∞$ does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of summability methods to produce convergence. The method of Cesàro summation leads to the Fejér kernel

$$F_N(x) = \frac1N\sum_{n=0}^{N-1} D_n(x) = \frac{1}{N}\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)^2.$$

The Fejér kernels tend to the delta function in a stronger sense that

$$\int_{-\pi}^{\pi} F_N(x)f(x)\,dx \to 2\pi f(0)$$

for every compactly supported function $f$. The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point.

Hilbert space theory
The Dirac delta distribution is a densely defined unbounded linear functional on the Hilbert space L2 of square-integrable functions. Indeed, smooth compactly supported functions are dense in $D_{N}$, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of $L^{2}$ and to give a stronger topology on which the delta function defines a bounded linear functional.

Sobolev spaces
The Sobolev embedding theorem for Sobolev spaces on the real line $L^{2}$ implies that any square-integrable function $[−π,π]$ such that

$$\|f\|_{H^1}^2 = \int_{-\infty}^\infty |\widehat{f}(\xi)|^2 (1+|\xi|^2)\,d\xi < \infty$$

is automatically continuous, and satisfies in particular

$$\delta[f]=|f(0)| < C \|f\|_{H^1}.$$

Thus $[−π,π]$ is a bounded linear functional on the Sobolev space $R$. Equivalently $f$ is an element of the continuous dual space $H^{1}$ of $H^{−1}$. More generally, in $[−π,π]$ dimensions, one has $H^{1}$ provided $δ ∈ H^{−s}(R^{n})$.

Spaces of holomorphic functions
In complex analysis, the delta function enters via Cauchy's integral formula, which asserts that if $f$ is a domain in the complex plane with smooth boundary, then

$$f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D$$

for all holomorphic functions $f$ in $δ$ that are continuous on the closure of $δ$. As a result, the delta function $s > n⁄2$ is represented in this class of holomorphic functions by the Cauchy integral:

$$\delta_z[f] = f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z}.$$

Moreover, let $δ_{z}$ be the Hardy space consisting of the closure in $H^{2}(∂D)$ of all holomorphic functions in $n$ continuous up to the boundary of $D$. Then functions in $L^{2}(∂D)$ uniquely extend to holomorphic functions in $f$, and the Cauchy integral formula continues to hold. In particular for $H^{2}(∂D)$, the delta function $D$ is a continuous linear functional on $z ∈ D$. This is a special case of the situation in several complex variables in which, for smooth domains $D$, the Szegő kernel plays the role of the Cauchy integral.

Resolutions of the identity
Given a complete orthonormal basis set of functions $H^{2}(∂D)$ in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector $D$ can be expressed as $$f = \sum_{n=1}^\infty \alpha_n \varphi_n. $$ The coefficients {αn} are found as $$\alpha_n = \langle \varphi_n, f \rangle,$$ which may be represented by the notation: $$\alpha_n = \varphi_n^\dagger f, $$ a form of the bra–ket notation of Dirac. Adopting this notation, the expansion of $D$ takes the dyadic form:

$$f = \sum_{n=1}^\infty \varphi_n \left ( \varphi_n^\dagger f \right). $$

Letting $D$ denote the identity operator on the Hilbert space, the expression

$$I = \sum_{n=1}^\infty \varphi_n \varphi_n^\dagger, $$

is called a resolution of the identity. When the Hilbert space is the space $\{φ_{n}\}$ of square-integrable functions on a domain $δ_{z}$, the quantity:

$$\varphi_n \varphi_n^\dagger, $$

is an integral operator, and the expression for $D$ can be rewritten

$$f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi.$$

The right-hand side converges to $f$ in the $L^{2}(D)$ sense. It need not hold in a pointwise sense, even when $f$ is a continuous function. Nevertheless, it is common to abuse notation and write

$$f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, $$

resulting in the representation of the delta function:

$$\delta(x-\xi) = \sum_{n=1}^\infty \varphi_n (x) \varphi_n^*(\xi). $$

With a suitable rigged Hilbert space $L^{2}$ where $(Φ, L^{2}(D), Φ*)$ contains all compactly supported smooth functions, this summation may converge in $Φ ⊂ L^{2}(D)$, depending on the properties of the basis $Φ*$. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the distribution sense.

Infinitesimal delta functions
Cauchy used an infinitesimal $I$ to write down a unit impulse, infinitely tall and narrow Dirac-type delta function $D$ satisfying $\int F(x)\delta_\alpha(x) \,dx = F(0)$ in a number of articles in 1827. Cauchy defined an infinitesimal in Cours d'Analyse (1827) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology.

Non-standard analysis allows one to rigorously treat infinitesimals. The article by contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals. Here the Dirac delta can be given by an actual function, having the property that for every real function $f$ one has $\int F(x)\delta_\alpha(x) \, dx = F(0)$ as anticipated by Fourier and Cauchy.

Dirac comb
A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Sha distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis. The Dirac comb is given as the infinite sum, whose limit is understood in the distribution sense,

$$\operatorname{\text{Ш}}(x) = \sum_{n=-\infty}^\infty \delta(x-n),$$

which is a sequence of point masses at each of the integers.

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if $f$ is any Schwartz function, then the periodization of $f$ is given by the convolution $$(f * \operatorname{\text{Ш}})(x) = \sum_{n=-\infty}^\infty f(x-n).$$ In particular, $$(f*\operatorname{\text{Ш}})^\wedge = \widehat{f}\widehat{\operatorname{\text{Ш}}} = \widehat{f}\operatorname{\text{Ш}}$$ is precisely the Poisson summation formula. More generally, this formula remains to be true if $α$ is a tempered distribution of rapid descent or, equivalently, if $$\widehat{f}$$ is a slowly growing, ordinary function within the space of tempered distributions.

Sokhotski–Plemelj theorem
The Sokhotski–Plemelj theorem, important in quantum mechanics, relates the delta function to the distribution $φ_{n}$, the Cauchy principal value of the function $p.v. 1⁄x$, defined by

$$\left\langle\operatorname{p.v.}\frac{1}{x}, \varphi\right\rangle = \lim_{\varepsilon\to 0^+}\int_{|x|>\varepsilon} \frac{\varphi(x)}{x}\,dx.$$

Sokhotsky's formula states that

$$\lim_{\varepsilon\to 0^+} \frac{1}{x\pm i\varepsilon} = \operatorname{p.v.}\frac{1}{x} \mp i\pi\delta(x),$$

Here the limit is understood in the distribution sense, that for all compactly supported smooth functions $δ_{α}$,

$$\int_{-\infty}^{\infty}\lim_{\varepsilon\to0^{+}}\frac{f(x)}{x\pm i\varepsilon}\,dx=\mp i\pi f(0)+\lim_{\varepsilon\to0^{+}}\int_{|x|>\varepsilon}\frac{f(x)}{x}\,dx.$$

Relationship to the Kronecker delta
The Kronecker delta $F$ is the quantity defined by

$$\delta_{ij} = \begin{cases} 1 & i=j\\ 0 &i\not=j \end{cases} $$

for all integers $T$, $f$. This function then satisfies the following analog of the sifting property: if $f$ (for $f$ in the set of all integers) is any doubly infinite sequence, then

$$\sum_{i=-\infty}^\infty a_i \delta_{ik}=a_k.$$

Similarly, for any real or complex valued continuous function $f$ on $1⁄x$, the Dirac delta satisfies the sifting property

$$\int_{-\infty}^\infty f(x)\delta(x-x_0)\,dx=f(x_0).$$

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.

Probability theory
In probability theory and statistics, the Dirac delta function is often used to represent a discrete distribution, or a partially discrete, partially continuous distribution, using a probability density function (which is normally used to represent absolutely continuous distributions). For example, the probability density function $R$ of a discrete distribution consisting of points $f(x)$, with corresponding probabilities $x = \{x_{1}, ..., x_{n}\}$, can be written as

$$f(x) = \sum_{i=1}^n p_i \delta(x-x_i).$$

As another example, consider a distribution in which 6/10 of the time returns a standard normal distribution, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution). The density function of this distribution can be written as

$$f(x) = 0.6 \, \frac {1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} + 0.4 \, \delta(x-3.5).$$

The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If $p_{1}, ..., p_{n}$ is a continuous differentiable function, then the density of $δ_{ij}$ can be written as

$$f_Y(y) = \int_{-\infty}^{+\infty} f_X(x) \delta(y-g(x)) \,dx. $$

The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion). The local time of a stochastic process $Y = g(X)$ is given by $$\ell(x,t) = \int_0^t \delta(x-B(s))\,ds$$ and represents the amount of time that the process spends at the point $i$ in the range of the process. More precisely, in one dimension this integral can be written $$\ell(x,t) = \lim_{\varepsilon\to 0^+}\frac{1}{2\varepsilon}\int_0^t \mathbf{1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds$$ where $$\mathbf{1}_{[x-\varepsilon,x+\varepsilon]}$$ is the indicator function of the interval $$[x-\varepsilon,x+\varepsilon].$$

Quantum mechanics
The delta function is expedient in quantum mechanics. The wave function of a particle gives the probability amplitude of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space $B(t)$ of square-integrable functions, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set $L^{2}$ of wave functions is orthonormal if

$$\langle\varphi_n \mid \varphi_m\rangle = \delta_{nm},$$

where $j$ is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function $\{|φ_{n}\rangle\}$ can be expressed as a linear combination of the $|ψ\rangle$ with complex coefficients:

$$ \psi = \sum c_n \varphi_n, $$

where $\{|φ_{n}\rangle\}$. Complete orthonormal systems of wave functions appear naturally as the eigenfunctions of the Hamiltonian (of a bound system) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the spectrum of the Hamiltonian. In bra–ket notation this equality implies the resolution of the identity:

$$I = \sum |\varphi_n\rangle\langle\varphi_n|.$$

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an observable can also be continuous. An example is the position operator, $ψ⟩$. The spectrum of the position (in one dimension) is the entire real line and is called a continuous spectrum. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well, i.e., to replace the Hilbert space with a rigged Hilbert space. In this context, the position operator has a complete set of "generalized eigenfunctions", labeled by the points $a_{i}$ of the real line, given by

$$\varphi_y(x) = \delta(x-y).$$

The eigenfunctions of the position operator are called the eigenstates and are denoted by $Qψ(x) = xψ(x)$.

Similar considerations apply to the eigenstates of the momentum operator, or indeed any other (unbounded) self-adjoint operator $i$ on the Hilbert space, provided the spectrum of $f$ is continuous and there are no degenerate eigenvalues. In that case, there is a set $φ_{y} = |y\rangle$ of real numbers (the spectrum) and a collection of distributions $Y$ with $Ω$ such that

$$P\varphi_y = y\varphi_y.$$

That is, $x$ are the generalized eigenvectors of $δ_{nm}$. If they form an "orthonormal basis" in the distribution sense, that is:

$$\langle \varphi_y,\varphi_{y'}\rangle = \delta(y-y'),$$

then for any test function $y$,

$$ \psi(x) = \int_\Omega c(y) \varphi_y(x) \, dy$$

where $y ∈ Ω$. That is, there is a resolution of the identity

$$I = \int_\Omega |\varphi_y\rangle\, \langle\varphi_y|\,dy$$

where the operator-valued integral is again understood in the weak sense. If the spectrum of $P$ has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum and an integral over the continuous spectrum.

The delta function also has many more specialized applications in quantum mechanics, such as the delta potential models for a single and double potential well.

Structural mechanics
The delta function can be used in structural mechanics to describe transient loads or point loads acting on structures. The governing equation of a simple mass–spring system excited by a sudden force impulse $P$ at time $c(y) = ⟨ψ, φ_{y}⟩$ can be written

$$m \frac{d^2 \xi}{dt^2} + k \xi = I \delta(t),$$

where $φ_{y}$ is the mass, $φ_{y}$ is the deflection, and $P$ is the spring constant.

As another example, the equation governing the static deflection of a slender beam is, according to Euler–Bernoulli theory,

$$EI \frac{d^4 w}{dx^4} = q(x),$$

where $ψ$ is the bending stiffness of the beam, $P$ is the deflection, $I$ is the spatial coordinate, and $t = 0$ is the load distribution. If a beam is loaded by a point force $m$ at $q(x)$, the load distribution is written

$$q(x) = F \delta(x-x_0).$$

As the integration of the delta function results in the Heaviside step function, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise polynomials.

Also, a point moment acting on a beam can be described by delta functions. Consider two opposing point forces $ξ$ at a distance $k$ apart. They then produce a moment $x = x_{0}$ acting on the beam. Now, let the distance $EI$ approach the limit zero, while $w$ is kept constant. The load distribution, assuming a clockwise moment acting at $M = Fd$, is written

$$\begin{align} q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\[4pt] &= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\[4pt] &= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\[4pt] &= M \delta'(x). \end{align}$$

Point moments can thus be represented by the derivative of the delta function. Integration of the beam equation again results in piecewise polynomial deflection.

title: Dirac delta function


text:
Courtesan

A courtesan is a prostitute with a courtly, wealthy, or upper-class clientele. Historically, the term referred to a courtier, a person who attended the court of a monarch or other powerful person.

History
In European feudal society, the court was the centre of government as well as the residence of the monarch, and social and political life were often completely mixed together. Prior to the Renaissance, courtesans served to convey information to visiting dignitaries, when servants could not be trusted. In Renaissance Europe, courtiers played an extremely important role in upper-class society. As it was customary during this time for royal couples to lead separate lives—commonly marrying simply to preserve bloodlines and to secure political alliances—men and women would often seek gratification and companionship from people living at court. In fact, the verb 'to court' originally meant "to be or reside at court", and later came to mean "to behave as a courtier" and then 'courtship', or "to pay amorous attention to somebody". The most intimate companion of a ruler was called the "favourite".

In Renaissance usage, the Italian word cortigiana, feminine of cortigiano ("courtier"), came to refer to a person who attends the court, and then to a well-educated and independent woman, eventually a trained artist or artisan of dance and singing, especially one associated with wealthy, powerful, or upper-class society who was given luxuries and status in exchange for entertainment and companionship. In English, the word was borrowed from Italian during the 16th century through the French form courtisane, especially associated with the meaning of donna di palazzo.

A male figure comparable to the courtesan was the Italian cicisbeo, the French chevalier servant, the Spanish cortejo or estrecho.

The courtesans of East Asia, particularly those of the Japanese empire, held a different social role than that of their European counterparts. Examples of Japanese courtesans included the oiran class, who were more focused on the aspect of entertainment than European courtesans.

Courtesans or dancers of ancient India known as ganikas were the center of city life. According to historian Sanjay K. Gautam, the courtesan in India was "a symbol of both sexual-erotic and aesthetic pleasure".

Categories
One type of courtesan was known (in Italy) as the cortigiana onesta, or the honest courtesan, who was cast as an intellectual. Another was the cortigiana di lume, a lower class of courtesan. The former was the sort most often romanticized and treated more-or-less equal to women of the nobility. It is with this type of courtesan that the art of "courtisanerie" is best associated.

The cortigiane oneste were usually well-educated and worldly (sometimes even more so than the average upper-class woman), and often held simultaneous careers as performers or artists. They were typically chosen on the basis of their "breeding"—social and conversational skills, intelligence, common sense, and companionship—as well as their physical attributes. It was usually their wit and personality that set them apart from regular women. Sex constituted only a facet of the courtesan's array of services. For example, they were well-dressed and ready to engage and participate in a variety of topics ranging from art to music to politics.

In some cases, courtesans were from well-to-do backgrounds, and were even married—but to husbands lower on the social ladder than their clients. In these cases, their relationships with those of high social status had the potential to improve their spouses' status—and so, more often than not, the husband was aware of his wife's profession and dealings.

As primary employment
Courtesans from non-wealthy backgrounds provided charming companionship for extended periods, no matter what their own feelings or commitments might have been at the time, and sometimes had to be prepared to do so on short notice. They were also subject to lower social status, and often religious disapproval, because of the perceived immoral aspects of their profession and their reliance upon courtisanerie as a primary source of income. In cases like this, a courtesan was solely dependent on her benefactor or benefactors financially, making her vulnerable; Cora Pearl is a good example. Often, courtesans serving in this capacity began their career as a prostitute, although many came to the profession by other means. It was not uncommon for a courtesan to enter into an arranged long-term liaison by contract with a wealthy benefactor. These contracts were written up by and witnessed by lawyers and were binding. Most included some provision for the financial welfare of the courtesan beyond the end of the relationship in the form of an annuity. Many such women became so powerful socially and financially that they could be particular about the men they associated with; in other words, they chose their paramour as would any other mistress, not the other way around. Wealthy benefactors would go to great lengths to court a courtesan as a prize, the ultimate goal being a long-term contract as a mistress.

Occasionally courtesans were passed from one benefactor to another, thereby resulting in them being viewed in society circles as lower than both their benefactor and those of wealth and power with whom they would socialise. Often, in instances of this sort, if the courtesan had satisfactorily served a benefactor, that benefactor would, when ending the affair, pass her on to another benefactor of wealth as a favour to the courtesan, or set her up in an arranged marriage to a semi-wealthy benefactor. If the courtesan had angered or dissatisfied a benefactor, then she would often find herself cast out of wealthy circles, returning more often than not to street prostitution.

For social or political benefits

 * Should not be confused with a royal mistress

Those from wealthy backgrounds, either by birth or marriage, and who were acting as courtesans only for the social or political advancement of themselves and/or their spouses were generally treated as equals. They were more respected by their extramarital companions, both placing one another's family obligations ahead of the relationship and planning their own liaisons or social engagements around the lovers' marital obligations.

Affairs of this sort would often be short-lived, ending when either the courtesan or the courtesan's spouse received the status or political position desired, or when the benefactor chose the company of another courtesan, and compensated the former companion financially. In instances like this, it was often viewed simply as a business agreement by both parties involved. The benefactor was aware of the political or social favors expected by the courtesan, the courtesan was aware of the price expected from them for those favors being carried out, and the two met one another's demands.

This was generally a safe affair, as both the benefactor's spouse and the courtesan's spouse usually were fully aware of the arrangement, and the courtesan was not solely dependent on the benefactor. It, rather, was simply an affair of benefits gained for both those involved. Publicly and socially, affairs of this sort were common during the 17th, 18th and 19th centuries, as well as the early 20th century, and were generally accepted in wealthy circles.

Career length

In later centuries, from the mid-18th century on, courtesans would often find themselves cast aside by their benefactors, but the days of public execution or imprisonment based on their promiscuous lifestyle were over. There are many examples of courtesans who, by remaining discreet and respectful to their benefactors, were able to extend their careers into or past middle age and retire financially secure; Catherine Walters is a good example. By the late 19th century, and for a brief period in the early 20th century, courtesans had reached a level of social acceptance in many circles and settings, often even to the extent of becoming a friend and confidant to the wife of their benefactor.

More often than not, a woman serving as a courtesan would last in that field only as long as she could prove herself useful to her companion, or companions. This, of course, excludes those who served as courtesans but who were already married into high society. When referring to those who made their service as a courtesan as their main source of income, success was based solely on financial management and longevity. Many climbed through the ranks of royalty, serving as mistress to lesser nobles first, eventually reaching the role of (unofficial) mistress to a king or prince.

Pietro Aretino, an Italian Renaissance writer, wrote a series of dialogues (Capricciosi ragionamenti) in which a mother teaches her daughter what options are available to women and how to be an effective courtesan. In eighteenth-century French literature, the courtesan tends to be idealised as a beautiful and intellectually refined figure of classical antiquity, who invariably enjoys excellent health and is untouched by the contemporary societal scourge of syphilis (often politely referred to as "smallpox") and other sexually transmitted diseases. The French novelist Balzac wrote about a courtesan in his Splendeurs et misères des courtisanes (1838–47). Émile Zola likewise wrote a novel, Nana (1880), about a courtesan in nineteenth-century France.

Famous courtesans
This is a list of some professional courtesans. They are not royal mistresses, unless a professional courtesan was also a royal mistress.

Separately from this list, the term "courtesan" has been used in a political context in an attempt to damage the reputation of a powerful woman or disparage her importance. Because of this, there is still much historical debate over whether certain women in history were courtesans. For example, the title was applied to the Byzantine empress Theodora, who had started life as an erotic actress but later became the wife of the Emperor Justinian and, after her death, an Orthodox saint.

17th century and earlier

 * Volumnia Cytheris (1st century BCE), Roman mimae actress and courtesan
 * Amrapali (5th century BCE), nagarvadhu dancer of Vaishali, following the Buddha's teachings she became an arahant.
 * Anarkali (17th-century) tawaif of Salim (later Mughal emperor Jahangir)
 * Arib al-Ma'muniyya (عَرِيب المأمونية, CE 797-890), qiyan
 * Faḍl al-Shāʻirah (فضل الشاعرة, d. 871 CE), qiyan
 * Hwang Jini (fl. 1550): legendary gisaeng of the Joseon Dynasty
 * ʽInān (عِنان, d. 841), qiyan
 * Imperia Cognati (1486–1512), courtesan of Renaissance Rome, referred to as the "first courtesan" in Europe
 * Kanhopatra (15th-century) Indian Marathi saint-poet and dancer,singer
 * Lais of Corinth (5th century BCE), hetaira-courtesan
 * Lais of Hyccara (killed 340 BCE), hetaira-courtesan
 * Marion Delorme (c. 1613–1650): lover of George Villiers, Duke of Buckingham, the Prince of Condé, and Cardinal Richelieu
 * Ninon de l'Enclos (1615–1705): lover of the Prince of Condé and Gaspard de Coligny
 * Phryne (4th century BCE), hetaira-courtesan
 * Praecia (fl. 73 BC), Roman courtesan
 * Shāriyah (شارِية, c. 815-870 CE), qiyan
 * Takao II (高尾, 1640–1659), Japanese oiran-courtesan
 * Thaïs (4th century BC), hetaira-courtesan
 * Theodora (6th century) (c. 500–June 28, 548), Byzantine actress-courtesan, later wife of Justinian I of the Byzantine Empire
 * List of Italian Renaissance courtesans:
 * Tullia d'Aragona (c. 1510–1556): top courtesan in several Italian cities, and published poet
 * Veronica Franco (1546–1591): a Venetian cortigiana onesta courtesan who was once lover to King Henry III of France and was depicted in the movie Dangerous Beauty
 * Yu Gam-dong (15th-century), Korean courtesan
 * Du Shiniang,China Ming Dynasty courtesan

18th and 19th centuries

 * Binodini Dasi (1862–1941), Indian tawaif-actress
 * Blanche d'Antigny (1840–1874), French courtesan; Émile Zola used her as the principal model for his novel Nana
 * Catherine Walters (1839–1920), British courtesan
 * Charlotte Slottsberg (1760–1800), Swedish courtesan-ballerina, lover but not official royal mistress of Charles XIII of Sweden
 * Claudine Guérin de Tencin (1681–1749), French courtesan and later a famous salonnière
 * Cora Pearl (1836–1886), demimonde-courtesan ('Grande Horizontale') of the Second Empire
 * The Countess, also known as Pauline, a French transgender courtesan and singer in Paris of the 1850s and 1860s
 * Dorothy Jordan (1761–1816), British courtesan-actress
 * Eliza Lynch (1835–1886), Irish courtesan, de facto wife of Francisco Solano López, president of Paraguay
 * Emma Hamilton (1765–1815), English model-actress, wife of William Hamilton and mistress of Lord Nelson
 * Grace Elliott (1754?–1823), British courtesan
 * Harriette Wilson (1786–1846), British courtesan
 * Josefa Ordóñez (1728 – d. after 1792), Mexican courtesan-actress
 * Kitty Fisher (died 1767), British courtesan and model
 * La Belle Otero (1868–1965), Spanish courtesan
 * La Païva (1819–1884), French demimonde-courtesan ('Grande Horizontale') of the Second Empire
 * Liane de Pougy (1869–1950), French courtesan and Folies Bergère-dancer
 * Lola Montez (1821–1861), Irish dancer, mistress of king Ludwig I of Bavaria
 * Madame du Barry (1743–1793), French courtesan, last Maîtresse-en-titre of Louis XV of France
 * Mah Laqa Bai (7 April 1768 – August 1824), Indian tawaif and poet
 * Marie Duplessis (1824–1847), French courtesan, one of the best known from the era of Louis Philippe
 * Marie-Louise O'Murphy (1737–1814), French courtesan, lover but not official royal mistress of Louis XV of France
 * Marthe de Florian (1864–1939), French courtesan and socialite
 * Mary Nesbitt (1742–1825), British courtesan and spy
 * Mata Hari (1876–1917), courtesan and spy
 * Olympe Pélissier (1799–1878), French courtesan and model, second wife of Gioachino Rossini
 * Rosalie Duthé (1748–1830), French courtesan, has been called "the first officially recorded dumb blonde"
 * Sai Jinhua (1872–1936), Chinese courtesan
 * Sophia Baddeley (1745–1786), British courtesan
 * Støvlet-Cathrine (1745–1805), Danish courtesan, lover but not official royal mistress of King Christian VII of Denmark
 * Zofia Potocka (1760–1822), Greek courtesan, mistress of Grigory Potemkin and wife of Szczęsny Potocki

20th century

 * Marguerite Alibert (1890–1971), French courtesan, lover but not official royal mistress of Prince Edward

In fiction

 * Angellica Bianca in Aphra Behn's 1677 play The Rover.
 * Bianca in William Shakespeare's Othello is considered a courtesan to Cassio.
 * Bianca, who appears in Anne Rice's The Vampire Armand, is a courtesan.
 * In John Cleland's Fanny Hill: or, the Memoirs of a Woman of Pleasure, Fanny goes from poor orphaned country girl to wealthy skilled courtesan eventually finding her one true love and retiring to marriage. Her history is told in the first person through several letters to friends detailing her life as a courtesan.
 * In Sarah Dunant's In the Company of the Courtesan, Fiammetta Bianchini, a renowned courtesan of Rome, and her sharp-witted dwarf rise to success among the intrigue and secrets of Renaissance Venice.
 * In the book A Great and Terrible Beauty, Pippa accuses Felicity of having a mother who is a courtesan and a consort, and who ran away to France not only to run a salon but to be with her lover, a Frenchman.
 * Inara Serra is a 26th-century Alliance companion, a position inspired by courtesans, in Joss Whedon's TV series Firefly.
 * Kamala, in Herman Hesse's Siddhartha.
 * Komagata Yumi in the manga Rurouni Kenshin.
 * La Dame aux Camélias is a novel about a courtesan by French author Alexandre Dumas, fils that was turned into the opera La Traviata by Italian composer Giuseppe Verdi. In the novel, the courtesan's name is Marguerite Gautier; in the opera, it is Violetta Valéry. "La Traviata" in Italian translates "The Wayward One".
 * Lysandra in the book series Throne of Glass by Sarah J. Maas, a shape-shifting courtesan working to pay off her debts and care for her rescued acolyte, Evangeline.
 * Madame Gabrielle from Dora Levy Mossanen. Courtesan: A Novel. Touchstone, 2005. ISBN 0-7432-4678-0
 * Magda in Puccini's La rondine.
 * Many examples in Indian literature and Bollywood films: tawaif Sahibjaan in Pakeezah, tawaif Umrao Jaan in the Urdu novel Umrao Jaan Ada and its adaptations, tawaif Chandramukhi in Devdas.
 * Mira Nair's 1996 film Kama Sutra: A Tale of Love highlights the profession of courtesans in 16th-century India, featuring Rasa Devi (Rekha) and Maya (Indira Varma).
 * Nana, in Emile Zola's eponymous novel of 1880 is a courtesan.
 * Odette de Crecy from Marcel Proust's In Search of Lost Time is a courtesan of the French Belle Epoque, she gains a notorious reputation from cavorting with aristocrats, artists and bourgeois, of both sexes.
 * Paola and Sister Teodora were the leaders of the courtesans of Florence and Venice (respectively) in the video game Assassin's Creed II. In its sequel, Assassin's Creed: Brotherhood, Madame Solari is shown to be the leader of the courtesans in Rome. Courtesans also provide a gameplay mechanic in the two games, main character Ezio Auditore can hire small groups of courtesans that can be used to escort the assassin without being noticed, and to distract hostile guards.
 * Phèdre nó Delaunay, the premier courtesan of Terre D'Ange in Jacqueline Carey's Kushiel's Legacy novels.
 * Satine, played by Nicole Kidman, an actress/courtesan who falls in love with a penniless poet/writer played by Ewan McGregor, in Baz Luhrmann's 2001 film, Moulin Rouge!.
 * Sha'ira, an asari "Consort" from the Mass Effect computer game series.
 * The Broadway plays, musicals, and movies based upon the book Gigi are about a young Parisian girl who is being trained to be a courtesan by her great-aunt, a retired career courtesan herself.
 * Ulla Winblad, in the famous 18th-century poems of Carl Michael Bellman.
 * Vasantasena, a nagarvadhu in the ancient Indian Sanskrit play Mṛcchakatika by Śūdraka.
 * Vittoria Corombona in John Webster's play The White Devil. She is described in the alternative title of the play as 'the famous Venetian Curtizan'.
 * In the movie Balls of Fury, Diedrich Bader plays one of Feng's many, all-male courtesans of pleasure.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Courtesan


text:
Conservatorship

Under U.S. law, a conservatorship results from the appointment of a guardian or a protector by a judge to manage the personal or financial affairs of another person who is incapable of fully managing their own affairs due to age or physical or mental limitations. A person under conservatorship is a "conservatee", a term that can refer to an adult. A person under guardianship is a "ward", a term that can also refer to a minor child. Conservatorship may also apply to corporations and organizations.

The conservator may be only of the "estate" (financial affairs) but may be also of the "person", wherein the conservator takes charge of overseeing the daily activities, such as healthcare or living arrangements of the conservatee. A conservator of the person is more typically called a legal guardian.

Appointment
Conservatorship is established either by court order (with regard to individuals) or via a statutory or regulatory authority (with regard to organizations such as business entities). In other legal terms, a conservatorship may refer to the legal responsibilities over a person who has a mental illness, including individuals who are psychotic, suicidal, demented, incapacitated, or in some other way unable to make legal, medical or financial decisions on behalf of themselves.

When referring to government control of private corporations such as Freddie Mac or Fannie Mae, conservatorship implies a more temporary control than does nationalisation.

Conservatorship of persons
Conservatorship is a legal term referring to the legal responsibilities of a conservator over the affairs of a person who has been deemed gravely disabled by the court and unable to meet their basic needs of food, clothing, and shelter. They are governed by the state's individual laws. Terminology varies, and some states or jurisdictions may refer to a conservator as a guardian of the estate or as a trustee.

Conservatorships are generally put in place for people who are significantly disabled by mental illness, elderly individuals who lack mental capacity due to medical conditions such as dementia, or individuals with developmental disabilities who lack the capacity to manage their own affairs. In typical conservatorship proceedings, an allegedly mentally incapacitated person must be evaluated by a qualified physician or psychiatrist who prepares a report documenting the person's mental capacity that is provided to the court and may be used as evidence.

Limited conservatorship
A "limited conservatorship" usually refers to the limited legal responsibilities of a conservator over the affairs of an individual who is developmentally disabled, but still capable of making important decisions for themselves. In these cases, the conservatee to whom the limited conservatorship applies can retain more control over their personal affairs than other conservatees can; for example, they may retain their right to decide where they may live.

An example of a conservator's duties includes: locating and marshalling assets, such as property and money, which belong to the conservatee; using the assets to buy food for the conservatee, secure and pay for placement in a facility which would take care of the conservatee or treat a mental illness, pay bills for the conservatee, manage property by paying for property insurance, mortgage payments or rent, property clean-up, or pay for a property management company to rent the property. An example of a conservator or guardian's medical responsibilities would be the court granting medical authority to the conservator or guardian, and the conservator or guardian authorising a physician to place a feeding tube to provide nourishment into the protected person's stomach if they are in medical need of it. It is not uncommon for one person to hold both offices and be referred to as the "guardian and conservator" of the conservatee, even though a conservator or guardian can be appointed over the person only, the estate only, or both. Generally, a conservator or guardian over the estate is only appointed if the conservatee has assets that need to be protected, marshalled, and managed. These terms may be found in use in Uniform Probate Code (UPC) jurisdictions, even though the UPC uses the term "protected person" in either case.

In most states, an outside party or agency must review the facts of the case and submit a report, usually required to be in writing, to the court before the court makes a decision on the request to establish a conservatorship or guardianship. Usually the outside party is a local county mental health representative called an investigator. They are often required to be experts in some appropriate field, such as social work, mental health, a medical field, or law. Procedures for conservatorship of an adult are often different from those for minors.

The court may appoint an attorney to represent the proposed conservatee or ward. If the proposed conservatee or ward is unable to have an attorney-client relationship because of some impairment, the court may appoint a guardian-ad-litem (who is often also an attorney). A guardian-ad-litem does not take instruction from the client, but rather acts on their behalf and tells the court what they think is in the best interests of the proposed conservatee or ward, whether or not that is what the proposed conservatee or ward wants. The conservatee has the right to be represented by an attorney, and if they cannot afford a private attorney, they are appointed a public defender that will represent them free of cost.

California
In the state of California there are two types of conservatorships: Lanterman–Petris–Short (Lanterman–Petris–Short Act of 1967, referred to as LPS) and Probate conservatorships. These forms of conservatorship are governed by the California Probate Code, and Welfare and Institutions Codes.

LPS conservatorships begin with a temporary 30-day conservatorship, and if the conservatee remains gravely disabled, the conservator is reappointed for a year; the LPS conservatorship can be renewed annually, or terminated if no longer needed. Probate conservatorships are referred to as "general conservatorships", and typically do not have a temporary period unless an urgent emergency exists that is creating risk to the person or their estate. Probate conservatorship do not automatically expire as LPS conservatorships do if they are not renewed by the conservator.

In an LPS conservatorship, a court-appointed conservator over the person is responsible for managing the conservatee's placement, medical decisions, and mental health treatment. A conservator over the estate is responsible for marshalling, protecting, and managing the conservatee's assets that remain in their estate. A conservator reports to the court that appointed them, and is monitored by the supervising judicial court in the county in which the conservatee permanently resides.

LPS conservatorships usually begin in the county mental health system and are referred from acute psychiatric hospitals, where Probate conservatorships can result from any referral source if validated with proper medical documentation. Mental Health consumers have the right to a Patient's Rights advocate, and are taken through a series of hearings while they are in the acute hospital before they reach the point of needing a conservator.

In 2022, a law was enacted requiring judges in California to document all alternatives to a conservatorship before granting one, giving potential conservatees in California preference in selecting a conservator, and making it easier to end probate conservatorships in California.

International equivalents
Conservatorship for individuals is called "deputyship" in England, Wales and in Switzerland (formerly "guardianship"), "guardianship" in Scotland, controllership in "Northern Ireland", "guardianship" in India and "guardianship and administration" in Australia.

South Korea
In the Republic of Korea (South Korea) a Conservatorship is called a Guardianship.

Types of Guardians under Korean Guardianship Law

Adult guardian (성년후견인): If an adult chronically lacks the mental competence to manage their own matters due to illness, disability, old age, or other conditions, a Korean court may appoint an adult guardian. This type of guardianship in Korea gives near total power over the ward to the Adult Guardian.

Limited guardian (한정후견인): A person may also be designated as a "special guardian," entrusted with restricted authority over the ward's interests. For example, a special guardian may be granted the legal authority in Korea to decide how to handle the ward's assets without being granted any control over the ward's person.

Specified guardian (특정후견인): A specified guardian is a person appointed to represent a person's interests in relation to a particular court proceeding or process.

The process of appointing a guardian through Korean courts
The Korean Family Courts, typically, has the authority to appoint a guardian in Korea. A general adult guardian is one who is in charge of both the ward's financial interests and personal welfare. The Korean family court, or one of its branches, has authority over the ward's address and will hear the guardianship case. When the Family Court is not present in the ward's address, typically, a district court or a branch court has jurisdiction over the matter.

Typically, after an evaluation of the ward's health by a doctor, the court proceedings begin. The court will often question the ward and hear his/her testimony regarding the guardianship. So that the ward can make the most use of his or her remaining capacity and choose a suitable guardian. The court has the power to decide the beginning of guardianship, the choice of a guardian, change of guardian, cessation of guardianship, the extent of the legal representative's authority, etc.

Conservatorship of organizations
In the United States, in some states, corporations can be placed under conservatorship, as a less extreme alternative to receivership. Whereas a receiver is expected to terminate the rights of shareholders and managers, a conservator is expected merely to assume those rights, with the prospect that they will be relinquished. Robert Ramsey and John Head, law professors who both specialise in financial issues, suggest that an insolvent bank should go into receivership rather than conservatorship to guard against false hope and moral hazard.

At the federal government level in the United States, in July 2008, the failing IndyMac Bank was taken into administrative receivership by the Federal Deposit Insurance Corporation (FDIC) and its assets and secured liabilities transferred to a specially established bridge bank called IndyMac Federal Bank, FSB which was placed into conservatorship, also by the FDIC.

Again, in the U.S. at the federal level, in September 2008, the chief executive officers and board of directors of Fannie Mae and of Freddie Mac were dismissed. Then, the companies were placed into the conservatorship of the Federal Housing Finance Agency (FHFA) via the determination of its director James B. Lockhart III, with the support and financial backing of U.S. Treasury via Treasury secretary Hank Paulson's commitment to keep the corporations solvent. The intervention leading to the conservatorship of these two entities has become the largest in government history, and was justified as necessary step to prevent the damage to the financial system that would have been caused by their failure. Entities like this are considered "too big to fail".

An even more ambitious use of the conservatorship model has been proposed by Duke professors Lawrence Baxter, Bill Brown, and Jim Cox. They suggest that the troubled U.S. banks be placed in conservatorship, that some of their "good assets" be dropped into newly created "good bank" subsidiaries (presumably under new management), and the remaining "bad assets" be left to be managed under the supervision of a conservatorship structure.

title: Conservatorship


text:
Falsificationism

Falsificationism may refer to:

 * Critical rationalism, an epistemological philosophy founded by Karl Popper
 * Three models of scientific progress in "Falsification and the Methodology of Scientific Research Programmes" by Imre Lakatos
 * Dogmatic falsificationism
 * Naive falsificationism
 * Sophisticated falsificationism

title: Falsificationism


text:
User:RK

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Experiment
TBA

Flint

Important thoughts

 * "Think of how stupid the average person is and realize that half of everybody is stupider than that." Comedian and writer George Carlin

 * "Nearly all men can stand adversity. But if you want to test a man's character - give him power." US President Abraham Lincoln

 * "Dante once said that the hottest places in hell are reserved for those who in a period of moral crisis maintain their neutrality."

 * US President John F. Kennedy, said in Bonn, West Germany, at the signing of a charter establishing the German Peace Corps, 1963. Public Papers of the Presidents of the United States: John F. Kennedy, 1963, p. 503. While Dante did not say this as such, JFK was paraphrasing Dante's quote from La Comedia Divina, canto 3, lines 35–42: “They are mixed with that repulsive choir of angels … undecided in neutrality. Heaven, to keep its beauty, cast them out, but even Hell itself would not receive them for fear the wicked there might glory over them.” Trans. Mark Musa, p. 21 (1971).

 * "...Everybody has opinions: I have them, you have them. And we are all told from the moment we open our eyes, that everyone is entitled to his or her opinion. Well, that's horsepuckey, of course. We are not entitled to our opinions; we are entitled to our informed opinions. Without research, without background, without understanding, it's nothing. It's just bibble-babble..." Writer Harlan Ellison

 * "We know nothing except through logical analysis, and if we reject that sole connexion with reality, we might as well stop trying to be adults and retreat into the capricious dream-world of infantility." H.P._Lovecraft, in a letter sent to Robert E. Howard, 8/16/1932

 * "Whoever preserves the life of a single human being ... it is as if he had preserved an entire world" Mishnah, Sanhedrin 4:5

On the role of scientists and those who do not value them

 * Our speculations on the growing antipathy toward academic radicalism on the part of scientists are influenced by a certain sense of how the humanities and the arts enter into the actual lives of our scientific colleagues. On the whole, scientists are deeply cultured people, in the best and most honorable sense. The image of the scientific monomaniac, of science departments devoted to a "naive scientism" is, to say the least, highly misleading. The range of knowledge of music, art, history, philosophy, and literature to be found in a random sample of scientists is, we know from long experience, extensive, and in some fortunate venues, enormous. Most of this learning has been acquired, of necessity, at odd moments here and there -- not through formal or systematic study. As humanists, therefore, scientists are autodidacts. One obvious consequence of this fact is to undercut the argument that traditional humanities departments, in their role as educators, are indispensable bearers of the great treasures of our cultural heritage. There are other, albeit less efficient, routes to education.

 * Let us be blunt: Having come so far, we have little left to lose. If, taking a fanciful hypothesis, the humanities department at MIT...were to walk out in a huff, the scientific faculty could, at need with enough released time, patch together a humanities curriculum, to be taught by the scientists themselves. It would have obvious gaps and rough spots, to be sure, and it might with some regularity prove inane; but on the whole it would be, we imagine, no worse than operative. What the opposite situation -- a walkout by the scientists -- would produce, as the humanities department tried to cope with the demand for science education, we leave to the reader's imagination.

 * This little exercise in oneupmanship is, of course, utter fantasy. But it does point to something real. The notion that scientists and engineers will always accept as axiomatic the competence and indispensability for higher education of humanists and social scientists is altogether too smug. Other sentiments are clearly astir. How these matters play out in American intellectual life will depend, to some degree, on the ability of the non-scientists to rein in the most grotesque tendencies in their respective fields.

 * Higher Superstition, Paul R. Gross and Norman Levitt, p.243-244.

A Liberal Defense of Zionism
The following quote is excerpted from a book by the physicist Steven Weinberg.

 * I write about Zionism as one who has no interest in the preservation of Judaism (or, I hasten to add, any other religion), but a great deal of interest in the preservation of Jews. There always were two different ways that Zionists viewed the return to Zion: as a duty for all Jews, imposed by their religion, or as an opportunity for those Jews who want or need to live in a Jewish nation. As an unreligious American Jew, I feel no desire or duty to change my nationality, and I am in no position to deplore the fact that few other American Jews want to become Israelis. But I very much hope that the land of Israel will continue to provide an opportunity for Jews to flee oppression in any country, as it has since 1948. This refuge was needed in the twentieth century far more than could ever have been expected in 1897, but because of Arab and British opposition it was tragically not available at the time of greatest need, in the 1930s and 1940s. Zionism today in part reflects the determination that Israel will be available as a refuge the next time it is needed.

 * Zionism also represents the intrusion - by purchase and settlement rather than conquest, at least until Arab assaults made military action necessary - of a democratic, scientifically sophisticated, secular culture into a part of the world that for centuries had been despotic, technically backward and obsessed with religion. For me, it is this essentially Western character of Zionism that gives it an attraction that goes beyond its defensive role. It will be betrayed if Orthodox zealots succeed in making Israel a theocratic state, but I can't believe that this will happen.

 * It is in part the Western character of Zionism that makes it so hateful to many others - not just to Muslims, but also to Westerners who for one reason or another find it comfortable to line up with the adversaries of Western civilisation. By attacking Zionism, it is very convenient to express solidarity with the poorer non-Western people of the world, without having actually to make sacrifices to help them. In this way, anti-Zionism has come to play the role of a gutter multiculturalism.

 * Of course, this is not the whole story. Anti-Zionism also serves to give vent to deep-seated anti-Semitic feelings, and helps in maintaining Middle Eastern oil. Out of this tangle of motivations came the hysterical anti-Zionism of the last decade of Stalinism; the resolution of a 1975 world congress on women at Mexico City that Zionism is oppressive to women; the infamous resolution of the General Assembly of the United Nations that same year equating Zionism with racism; and the continuing one-sidedness of international opinion regarding the relations between Israel and the Palestinian Arabs.

 * As a liberal I especially deplore a peculiar evenhandedness on the part of many fellow liberals, an evenhandedness that sees no moral difference between the efforts of Israel to preserve existence in a fraction of a fraction of the original Palestine mandate, and the attacks of those who wish to destroy it, and that views the building of housing projects in Jerusalem as being on a moral plane with the firing of machine guns at school buses. Do I need to say that there is in fact a great moral distinction between the democratic secular Israel created by Zionism, whose long-range aim is simply to be left in peace, and the enemies that surround it? It is a distinction that ought to make liberals see Zionism again as they used to, as a natural part of the liberal agenda.

Source: Steven Weinberg, “Facing Up,” Harvard University Press, Cambridge, Massachusetts, 2001, p. 182.

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