Dirac delta function

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Dirac_delta_function

Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.

The Dirac delta as the limit as ${\displaystyle a\to 0}$ (in the sense of distributions) of the sequence of zero-centered normal distributions ${\displaystyle {\delta }_a(x)=\frac{1}{\left|a\right|\sqrt{\pi }}{e}^{-(x/a{)}^2}}$

In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.

The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., ${\displaystyle f(x)}$) to its value at zero of its domain (${\displaystyle f(0)}$), or as the weak limit of a sequence of bump functions (e.g., ${\displaystyle \delta (x)=\underset{b\to 0}{lim}\frac{1}{|b|\sqrt{\pi }}{e}^{-(x/b{)}^2}}$), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions.

The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on functions.

The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is the discrete analog of the Dirac delta function.

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 ${\displaystyle t=0}$ 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 ${\displaystyle \mathrm{\Delta }t=[0,T]}$. That is,

$${\displaystyle F_{\mathrm{\Delta }t}(t)=\{\begin{array}{ll}P/\mathrm{\Delta }t& 0<t\le T,\\ 0& \text{otherwise}.\end{array}}$$

Then the momentum at any time t is found by integration:

$${\displaystyle p(t)={\int }_0^t{F}_{\mathrm{\Delta }t}(\tau )d\tau =\{\begin{array}{ll}P& t\ge T\\ Pt/\mathrm{\Delta }t& 0\le t\le T\\ 0& \text{otherwise.}\end{array}}$$

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δt → 0, giving a result everywhere except at 0:

$${\displaystyle p(t)=\{\begin{array}{ll}P& t>0\\ 0& t<0.\end{array}}$$

Here the functions ${\displaystyle F_{\mathrm{\Delta }t}}$ 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) $\underset{\mathrm{\Delta }t\to 0}{lim}{F}_{\mathrm{\Delta }t}$ 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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{F}_{\mathrm{\Delta }t}(t)dt=P,}$$

which holds for all ${\displaystyle \mathrm{\Delta }t>0}$, should continue to hold in the limit. So, in the equation $F(t)=P\delta (t)=\underset{\mathrm{\Delta }t\to 0}{lim}{F}_{\mathrm{\Delta }t}(t)$, it is understood that the limit is always taken outside the integral.

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:

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

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

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

Later, Augustin Cauchy expressed the theorem using exponentials:

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

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

$${\displaystyle \begin{array}{rl}f(x)& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{ipx}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-ip\alpha }f(\alpha )d\alpha \right)dp\\ & =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{ipx}{e}^{-ip\alpha }dp\right)f(\alpha )d\alpha ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x-\alpha )f(\alpha )d\alpha ,\end{array}}$$

where the δ-function is expressed as

$${\displaystyle \delta (x-\alpha )=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{ip(x-\alpha )}dp.}$$

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

$${\displaystyle \delta (x)\simeq \{\begin{array}{ll}+\mathrm{\infty },& x=0\\ 0,& x\ne 0\end{array}}$$

and which is also constrained to satisfy the identity

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)dx=1.}$$

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

Another equivalent definition of the Dirac delta function: ${\displaystyle \delta (x)}$ is a function (in a loose sense) that satisfies

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)dx=1,{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)g(x)dx=g(0)}$$

where g(x) is a well-behaved function. The second condition in this definition can be derived by the first definition above:

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)g(x)dx=\underset{ϵ\to 0}{lim}{\int }_{-ϵ}^{ϵ}\delta (x)g(x)dx=g(0)\underset{ϵ\to 0}{lim}{\int }_{-ϵ}^{ϵ}\delta (x)dx=g(0).}$$

The Dirac delta function can be rigorously defined either as a distribution or as a measure as described below.

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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\delta (dx)=f(0)}$$

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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\delta (x)dx=f(0)}$$

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.

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

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

$${\displaystyle H(x)={\int }_{\mathbf{R}}{\mathbf{1}}_{(-\mathrm{\infty },x]}(t)\delta (dt)=\delta (-\mathrm{\infty },x],}$$

the latter being the measure of this interval; more formally, δ((−∞, x]). Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\delta (dx)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dH(x).}$$

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 about 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

${\displaystyle \delta [\phi ]=\phi (0)}$

(1)

for every test function φ.

For δ 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 S on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer N there is an integer M_N and a constant C_N such that for every test function φ, one has the inequality

$${\displaystyle \left|S[\phi ]\right|\le C_N\sum _{k=0}^{M_N}\underset{x\in [-N,N]}{sup}\left|{\phi }^{(k)}(x)\right|}$$

where sup represents the supremum. With the δ distribution, one has such an inequality (with C_N = 1) with M_N = 0 for all N. Thus δ 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 φ, one has

$${\displaystyle \delta [\phi ]=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\phi }^{\prime }(x)H(x)dx.}$$

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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\phi (x)H^{\prime }(x)dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\phi (x)\delta (x)dx,}$$

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

$${\displaystyle -{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\phi }^{\prime }(x)H(x)dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\phi (x)dH(x).}$$

In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation (1) defines a Daniell integral on the space of all compactly supported continuous functions φ which, by the Riesz representation theorem, can be represented as the Lebesgue integral of φ concerning 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 n-dimensional Euclidean space Rⁿ as the measure such that

$${\displaystyle {\int }_{{\mathbf{R}}^n}f(\mathbf{x})\delta (d\mathbf{x})=f(\mathbf{0})}$$

for every compactly supported continuous function f. As a measure, the n-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with x = (x₁, x₂, ..., x_n), one has

${\displaystyle \delta (\mathbf{x})=\delta (x_1)\delta (x_2)\cdots \delta (x_n).}$

(2)

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, (2) 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 X is a set, x₀ ∈ X is a marked point, and Σ is any sigma algebra of subsets of X, then the measure defined on sets A ∈ Σ by

$${\displaystyle {\delta }_{x_0}(A)=\{\begin{array}{ll}1& \text{if }{x}_0\in A\\ 0& \text{if }{x}_0\notin A\end{array}}$$

is the delta measure or unit mass concentrated at x₀.

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 M centered at the point x₀ ∈ M is defined as the following distribution:

${\displaystyle {\delta }_{x_0}[\phi ]=\phi (x_0)}$

(3)

for all compactly supported smooth real-valued functions φ on M. A common special case of this construction is a case in which M is an open set in the Euclidean space Rⁿ.

On a locally compact Hausdorff space X, the Dirac delta measure concentrated at a point x is the Radon measure associated with the Daniell integral (3) on compactly supported continuous functions φ. 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 ${\displaystyle x_0\mapsto {\delta }_{x_0}}$ is a continuous embedding of X into the space of finite Radon measures on X, equipped with its vague topology. Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X.

Properties

Scaling and symmetry

The delta function satisfies the following scaling property for a non-zero scalar α:

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\alpha x)dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (u)\frac{du}{|\alpha |}=\frac{1}{|\alpha |}}$$

and so

${\displaystyle \delta (\alpha x)=\frac{\delta (x)}{|\alpha |}.}$

(4)

Scaling property proof:

$${\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dxg(x)\delta (ax)=\frac{1}{a}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}d{x}^{\prime }g\left(\frac{x^{\prime }}{a}\right)\delta (x^{\prime })}$$

where a change of variable x′ = ax is used. If a is negative, i.e., a = −|a|, then

$${\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dxg(x)\delta (ax)=\frac{1}{-\left|a\right|}\underset{\mathrm{\infty }}{\overset{-\mathrm{\infty }}{\int }}d{x}^{\prime }g\left(\frac{x^{\prime }}{a}\right)\delta (x^{\prime })=\frac{1}{\left|a\right|}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}d{x}^{\prime }g\left(\frac{x^{\prime }}{a}\right)\delta (x^{\prime })=\frac{1}{\left|a\right|}g(0).}$$

Thus, ${\displaystyle \delta (ax)=\frac{1}{\left|a\right|}\delta (x)}$.

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

$${\displaystyle \delta (-x)=\delta (x)}$$

which is homogeneous of degree −1.

Algebraic properties

The distributional product of δ with x is equal to zero:

$${\displaystyle x\delta (x)=0.}$$

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

Conversely, if xf(x) = xg(x), where f and g are distributions, then

$${\displaystyle f(x)=g(x)+c\delta (x)}$$

for some constant c.

Translation

The integral of the time-delayed Dirac delta is

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\delta (t-T)dt=f(T).}$$

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 ${\displaystyle {\delta }_T(t)=\delta (t-T)}$ is to time-delay f(t) by the same amount:

$${\displaystyle \begin{array}{rl}(f\ast {\delta }_T)(t)& \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )\delta (t-T-\tau )d\tau \\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )\delta (\tau -(t-T))d\tau \text{since}\delta (-x)=\delta (x)\text{by (4)}\\ & =f(t-T).\end{array}}$$

The sifting property holds under the precise condition that f 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)

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\xi -x)\delta (x-\eta )dx=\delta (\eta -\xi ).}$$

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

$${\displaystyle {\int }_{\mathbb{R}}\delta (g(x))f(g(x))\left|g^{\prime }(x)\right|dx={\int }_{g(\mathbb{R})}\delta (u)f(u)du}$$

provided that g is a continuously differentiable function with g′ nowhere zero. That is, there is a unique way to assign meaning to the distribution ${\displaystyle \delta \circ g}$ so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g′ = 0 point. This distribution satisfies δ(g(x)) = 0 if g is nowhere zero, and otherwise if g has a real root at x₀, then

$${\displaystyle \delta (g(x))=\frac{\delta (x-x_0)}{|g^{\prime }(x_0)|}.}$$

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

$${\displaystyle \delta (g(x))=\sum _i\frac{\delta (x-x_i)}{|g^{\prime }(x_i)|}}$$

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

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

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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\delta (g(x))dx=\sum _i\frac{f(x_i)}{|g^{\prime }(x_i)|}.}$$

Indefinite integral

For a constant ${\displaystyle a\in \mathbb{R}}$ and a "well-behaved" arbitrary real-valued function y(x),

$${\displaystyle {\displaystyle \int y(x)\delta (x-a)dx=y(a)H(x-a)+c,}}$$

where H(x) is the Heaviside step function and c is an integration constant.

Properties in n dimensions

The delta distribution in an n-dimensional space satisfies the following scaling property instead,

$${\displaystyle \delta (\alpha \mathbf{x})=|\alpha {|}^{-n}\delta (\mathbf{x}),}$$

so that δ is a homogeneous distribution of degree −n.

Under any reflection or rotation ρ, the delta function is invariant,

$${\displaystyle \delta (\rho \mathbf{x})=\delta (\mathbf{x}).}$$

As in the one-variable case, it is possible to define the composition of δ with a bi-Lipschitz function g: Rⁿ → Rⁿ uniquely so that the identity

$${\displaystyle {\int }_{{\mathbb{R}}^n}\delta (g(\mathbf{x}))f(g(\mathbf{x}))\left|det{g}^{\prime }(\mathbf{x})\right|d\mathbf{x}={\int }_{g({\mathbb{R}}^n)}\delta (\mathbf{u})f(\mathbf{u})d\mathbf{u}}$$

for all compactly supported functions f.

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 : Rⁿ → R such that the gradient of g is nowhere zero, the following identity holds

$${\displaystyle {\int }_{{\mathbb{R}}^n}f(\mathbf{x})\delta (g(\mathbf{x}))d\mathbf{x}={\int }_{g^{-1}(0)}\frac{f(\mathbf{x})}{|\mathrm{\nabla }g|}d\sigma (\mathbf{x})}$$

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 S is a smooth hypersurface of Rⁿ, then we can associate to S the distribution that integrates any compactly supported smooth function g over S:

$${\displaystyle {\delta }_S[g]={\int }_{S}g(\mathbf{s})d\sigma (\mathbf{s})}$$

where σ is the hypersurface measure associated to S. This generalization is associated with the potential theory of simple layer potentials on S. If D is a domain in Rⁿ with smooth boundary S, then δ_S is equal to the normal derivative of the indicator function of D in the distribution sense,

$${\displaystyle -{\int }_{{\mathbb{R}}^n}g(\mathbf{x})\frac{\mathrm{\partial }{1}_D(\mathbf{x})}{\mathrm{\partial }n}d\mathbf{x}={\int }_{S}g(\mathbf{s})d\sigma (\mathbf{s}),}$$

where n 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:

$${\displaystyle \delta (\mathbf{r}-{\mathbf{r}}_0)=\{\begin{array}{ll}{\displaystyle \frac{1}{r^2\sin \theta }\delta (r-r_0)\delta (\theta -{\theta }_0)\delta (\varphi -{\varphi }_0)}& x_0,y_0,z_0\ne 0\\ {\displaystyle \frac{1}{2\pi r^2\sin \theta }\delta (r-r_0)\delta (\theta -{\theta }_0)}& x_0=y_0=0,z_0\ne 0\\ {\displaystyle \frac{1}{4\pi r^2}\delta (r-r_0)}& x_0=y_0=z_0=0\end{array}}$$

Fourier transform

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

$${\displaystyle \hat{\delta }(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-2\pi ix\xi }\delta (x)dx=1.}$$

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

$${\displaystyle ⟨\hat{\delta },\phi ⟩=⟨\delta ,\hat{\phi }⟩}$$

for all Schwartz functions φ. And indeed it follows from this that ${\displaystyle \hat{\delta }=1.}$

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

$${\displaystyle S\ast \delta =S.}$$

That is to say that δ 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 δ, and once it is known, it characterizes the system completely. See LTI system theory § Impulse response and convolution.

The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed as

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}1\cdot e^{2\pi ix\xi }d\xi =\delta (x)}$$

and more rigorously, it follows since

$${\displaystyle ⟨1,\hat{f}⟩=f(0)=⟨\delta ,f⟩}$$

for all Schwartz functions f.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{i2\pi {\xi }_{1}t}{\left[e^{i2\pi {\xi }_{2}t}\right]}^{\ast }dt={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i2\pi ({\xi }_2-{\xi }_1)t}dt=\delta ({\xi }_2-{\xi }_1).}$$

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution

$${\displaystyle f(t)=e^{i2\pi {\xi }_{1}t}}$$

is

$${\displaystyle \hat{f}({\xi }_2)=\delta ({\xi }_1-{\xi }_2)}$$

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^[49]

$${\displaystyle {\int }_0^{\mathrm{\infty }}\delta (t-a)e^{-st}dt=e^{-sa}.}$$

Derivatives of the Dirac delta function

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 φ by

$${\displaystyle {\delta }^{\prime }[\phi ]=-\delta [{\phi }^{\prime }]=-{\phi }^{\prime }(0).}$$

The first equality here is a kind of integration by parts, for if δ were a true function then

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\delta }^{\prime }(x)\phi (x)dx=\delta (x)\phi (x){|}_{-\mathrm{\infty }}^{\mathrm{\infty }}-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x){\phi }^{\prime }(x)dx=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x){\phi }^{\prime }(x)dx=-{\phi }^{\prime }(0).}$$

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

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

In particular, δ is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:

$${\displaystyle {\delta }^{\prime }(x)=\underset{h\to 0}{lim}\frac{\delta (x+h)-\delta (x)}{h}.}$$

More properly, one has

$${\displaystyle {\delta }^{\prime }=\underset{h\to 0}{lim}\frac{1}{h}({\tau }_h\delta -\delta )}$$

where τ_h is the translation operator, defined on functions by τ_hφ(x) = φ(x + h), and on a distribution S by

$${\displaystyle ({\tau }_{h}S)[\phi ]=S[{\tau }_{-h}\phi ].}$$

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:

$${\displaystyle \begin{array}{rl}{\delta }^{\prime }(-x)& =-{\delta }^{\prime }(x)\\ x{\delta }^{\prime }(x)& =-\delta (x)\end{array}}$$

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, Liebnitz's theorem and linearity of inner product:

$${\displaystyle \begin{array}{rl}⟨x{\delta }^{\prime },\phi ⟩& =⟨{\delta }^{\prime },x\phi ⟩=-⟨\delta ,(x\phi {)}^{\prime }⟩=-⟨\delta ,x^{\prime }\phi +x{\phi }^{\prime }⟩=-⟨\delta ,x^{\prime }\phi ⟩-⟨\delta ,x{\phi }^{\prime }⟩=-x^{\prime }(0)\phi (0)-x(0){\phi }^{\prime }(0)\\ & =-x^{\prime }(0)⟨\delta ,\phi ⟩-x(0)⟨\delta ,{\phi }^{\prime }⟩=-x^{\prime }(0)⟨\delta ,\phi ⟩+x(0)⟨{\delta }^{\prime },\phi ⟩=⟨x(0){\delta }^{\prime }-x^{\prime }(0)\delta ,\phi ⟩\\ ⟹x(t){\delta }^{\prime }(t)& =x(0){\delta }^{\prime }(t)-x^{\prime }(0)\delta (t)=-x^{\prime }(0)\delta (t)=-\delta (t)\end{array}}$$

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

$${\displaystyle {\delta }^{\prime }\ast f=\delta \ast f^{\prime }=f^{\prime },}$$

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

Higher dimensions

More generally, on an open set U in the n-dimensional Euclidean space ${\displaystyle {\mathbb{R}}^n}$, the Dirac delta distribution centered at a point a ∈ U is defined by

$${\displaystyle {\delta }_a[\phi ]=\phi (a)}$$

for all ${\displaystyle \phi \in C_c^{\mathrm{\infty }}(U)}$, the space of all smooth functions with compact support on U. If ${\displaystyle \alpha =({\alpha }_1,\dots ,{\alpha }_n)}$ is any multi-index with ${\displaystyle |\alpha |={\alpha }_1+\cdots +{\alpha }_n}$ and ${\displaystyle {\mathrm{\partial }}^{\alpha }}$ denotes the associated mixed partial derivative operator, then the α-th derivative ∂^αδ_a of δ_a is given by

$${\displaystyle ⟨{\mathrm{\partial }}^{\alpha }{\delta }_a,\phi ⟩=(-1{)}^{|\alpha |}⟨{\delta }_a,{\mathrm{\partial }}^{\alpha }\phi ⟩=(-1{)}^{|\alpha |}{\mathrm{\partial }}^{\alpha }\phi (x){|}_{x=a}\text{ for all }\phi \in C_c^{\mathrm{\infty }}(U).}$$

That is, the α-th derivative of δ_a is the distribution whose value on any test function φ is the α-th derivative of φ at 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 S is any distribution on U supported on the set {a} consisting of a single point, then there is an integer m and coefficients c_α such that

$${\displaystyle S=\sum _{|\alpha |\le m}{c}_{\alpha }{\mathrm{\partial }}^{\alpha }{\delta }_a.}$$

Representations of the delta function

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

$${\displaystyle \delta (x)=\underset{\epsilon \to 0^{+}}{lim}{\eta }_{\epsilon }(x),}$$

where η_ε(x) is sometimes called a nascent delta function. This limit is meant in a weak sense: either that

${\displaystyle \underset{\epsilon \to 0^{+}}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\eta }_{\epsilon }(x)f(x)dx=f(0)}$

(5)

for all continuous functions f having compact support, or that this limit holds for all smooth functions f 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 η_ε can be constructed in the following manner. Let η be an absolutely integrable function on R of total integral 1, and define

$${\displaystyle {\eta }_{\epsilon }(x)={\epsilon }^{-1}\eta \left(\frac{x}{\epsilon }\right).}$$

In n dimensions, one uses instead the scaling

$${\displaystyle {\eta }_{\epsilon }(x)={\epsilon }^{-n}\eta \left(\frac{x}{\epsilon }\right).}$$

Then a simple change of variables shows that η_ε also has integral 1. One may show that (5) holds for all continuous compactly supported functions f, and so η_ε converges weakly to δ 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 L¹(R) of absolutely integrable functions is closed under the operation of convolution of functions: f ∗ g ∈ L¹(R) whenever f and g are in L¹(R). However, there is no identity in L¹(R) for the convolution product: no element h such that f ∗ h = f for all f. Nevertheless, the sequence η_ε does approximate such an identity in the sense that

$${\displaystyle f\ast {\eta }_{\epsilon }\to f\text{as }\epsilon \to 0.}$$

This limit holds in the sense of mean convergence (convergence in L¹). Further conditions on the η_ε, 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 = η₁ is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing η to be a suitably normalized bump function, for instance

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

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η₁ to be a hat function. With this choice of η₁, one has

$${\displaystyle {\eta }_{\epsilon }(x)={\epsilon }^{-1}max\left(1-\left|\frac{x}{\epsilon }\right|,0\right)}$$

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 η₁ 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 η₁ to be any probability distribution at all, and letting η_ε(x) = η₁(x/ε)/ε as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments. For instance, if η₁ is the uniform distribution on $\left[-\frac{1}{2},\frac{1}{2}\right]$, also known as the rectangular function, then:

$${\displaystyle {\eta }_{\epsilon }(x)=\frac{1}{\epsilon }\mathrm{rect}\left(\frac{x}{\epsilon }\right)=\{\begin{array}{ll}\frac{1}{\epsilon },& -\frac{\epsilon }{2}<x<\frac{\epsilon }{2},\\ 0,& \text{otherwise}.\end{array}}$$

Another example is with the Wigner semicircle distribution

$${\displaystyle {\eta }_{\epsilon }(x)=\{\begin{array}{ll}\frac{2}{\pi {\epsilon }^2}\sqrt{{\epsilon }^2-x^2},& -\epsilon <x<\epsilon ,\\ 0,& \text{otherwise}.\end{array}}$$

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 η_ε with η_δ must satisfy

$${\displaystyle {\eta }_{\epsilon }\ast {\eta }_{\delta }={\eta }_{\epsilon +\delta }}$$

for all ε, δ > 0. Convolution semigroups in L¹ 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

$${\displaystyle \{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}\eta (t,x)=A\eta (t,x),t>0\\ {\displaystyle \underset{t\to 0^{+}}{lim}\eta (t,x)=\delta (x)}\end{array}}$$

in which the limit is as usual understood in the weak sense. Setting η_ε(x) = η(ε, x) 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

$${\displaystyle {\eta }_{\epsilon }(x)=\frac{1}{\sqrt{2\pi \epsilon }}{\mathrm{e}}^{-\frac{x^2}{2\epsilon }}}$$

represents the temperature in an infinite wire at time t > 0, 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:

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

In probability theory, η_ε(x) is a normal distribution of variance ε and mean 0. It represents the probability density at time t = ε 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 Rⁿ, the heat kernel is

$${\displaystyle {\eta }_{\epsilon }=\frac{1}{(2\pi \epsilon {)}^{n/2}}{\mathrm{e}}^{-\frac{x\cdot x}{2\epsilon }},}$$

and has the same physical interpretation, mutatis mutandis. It also represents a nascent delta function in the sense that η_ε → δ in the distribution sense as ε → 0.

The Poisson kernel

The Poisson kernel

$${\displaystyle {\eta }_{\epsilon }(x)=\frac{1}{\pi }\mathrm{I}\mathrm{m}\left\{\frac{1}{x-\mathrm{i}\epsilon }\right\}=\frac{1}{\pi }\frac{\epsilon }{{\epsilon }^2+x^2}=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}\xi x-|\epsilon \xi |}d\xi }$$

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

$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}=-{\left(-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)}^{\frac{1}{2}}u(t,x)}$$

where the operator is rigorously defined as the Fourier multiplier

$${\displaystyle \mathcal{F}\left[{\left(-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)}^{\frac{1}{2}}f\right](\xi )=|2\pi \xi |\mathcal{F}f(\xi ).}$$

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

$${\displaystyle {\epsilon }^{-1/3}\mathrm{Ai}\left(x{\epsilon }^{-1/3}\right).}$$

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 R¹⁺¹:

$${\displaystyle \begin{array}{rl}{c}^{-2}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}-\mathrm{\Delta }u& =0\\ u=0,\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\delta & \text{for }t=0.\end{array}}$$

The solution u 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)

$${\displaystyle {\eta }_{\epsilon }(x)=\frac{1}{\pi x}\sin \left(\frac{x}{\epsilon }\right)=\frac{1}{2\pi }{\int }_{-\frac{1}{\epsilon }}^{\frac{1}{\epsilon }}\cos (kx)dk}$$

and the Bessel function

$${\displaystyle {\eta }_{\epsilon }(x)=\frac{1}{\epsilon }{J}_{\frac{1}{\epsilon }}\left(\frac{x+1}{\epsilon }\right).}$$

Plane wave decomposition

One approach to the study of a linear partial differential equation

$${\displaystyle L[u]=f,}$$

where L is a differential operator on Rⁿ, is to seek first a fundamental solution, which is a solution of the equation

$${\displaystyle L[u]=\delta .}$$

When L 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

$${\displaystyle L[u]=h}$$

where h is a plane wave function, meaning that it has the form

$${\displaystyle h=h(x\cdot \xi )}$$

for some vector ξ. Such an equation can be resolved (if the coefficients of L 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 k so that n + k is an even integer, and for a real number s, put

$${\displaystyle g(s)=\mathrm{Re}\left[\frac{-s^k\log (-is)}{k!(2\pi i{)}^n}\right]=\{\begin{array}{ll}\frac{|s{|}^k}{4k!(2\pi i{)}^{n-1}}& n\text{ odd}\\ -\frac{|s{|}^k\log |s|}{k!(2\pi i{)}^n}& n\text{ even.}\end{array}}$$

Then δ is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure dω of g(x · ξ) for ξ in the unit sphere Sⁿ⁻¹:

$${\displaystyle \delta (x)={\mathrm{\Delta }}_x^{(n+k)/2}{\int }_{S^{n-1}}g(x\cdot \xi )d{\omega }_{\xi }.}$$

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ,

$${\displaystyle \phi (x)={\int }_{{\mathbf{R}}^n}\phi (y)dy{\mathrm{\Delta }}_x^{\frac{n+k}{2}}{\int }_{S^{n-1}}g((x-y)\cdot \xi )d{\omega }_{\xi }.}$$

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 φ(x) from its integrals over hyperplanes. For instance, if n is odd and k = 1, then the integral on the right hand side is

$${\displaystyle \begin{array}{rl}& c_n{\mathrm{\Delta }}_x^{\frac{n+1}{2}}{\iint }_{S^{n-1}}\phi (y)|(y-x)\cdot \xi |d{\omega }_{\xi }dy\\ & =c_n{\mathrm{\Delta }}_x^{(n+1)/2}{\int }_{S^{n-1}}d{\omega }_{\xi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|p|R\phi (\xi ,p+x\cdot \xi )dp\end{array}}$$

where Rφ(ξ, p) is the Radon transform of φ:

$${\displaystyle R\phi (\xi ,p)={\int }_{x\cdot \xi =p}f(x)d^{n-1}x.}$$

An alternative equivalent expression of the plane wave decomposition is:

$${\displaystyle \delta (x)=\{\begin{array}{ll}\frac{(n-1)!}{(2\pi i{)}^n}{\displaystyle {\int }_{S^{n-1}}(x\cdot \xi {)}^{-n}d{\omega }_{\xi }}& n\text{ even}\\ \frac{1}{2(2\pi i{)}^{n-1}}{\displaystyle {\int }_{S^{n-1}}{\delta }^{(n-1)}(x\cdot \xi )d{\omega }_{\xi }}& n\text{ odd}.\end{array}}$$

Fourier kernels

See also: Convergence of Fourier series

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 n-th partial sum of the Fourier series of a function f of period 2π is defined by convolution (on the interval [−π,π]) with the Dirichlet kernel:

$${\displaystyle D_N(x)=\sum _{n=-N}^N{e}^{inx}=\frac{\sin \left(\left(N+\frac{1}{2}\right)x\right)}{\sin (x/2)}.}$$

Thus,

$${\displaystyle s_N(f)(x)=D_N\ast f(x)=\sum _{n=-N}^N{a}_n{e}^{inx}}$$

where

$${\displaystyle a_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(y)e^{-iny}dy.}$$

A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval [−π,π] tends to a multiple of the delta function as N → ∞. This is interpreted in the distribution sense, that

$${\displaystyle s_N(f)(0)={\int }_{-\pi }^{\pi }{D}_N(x)f(x)dx\to 2\pi f(0)}$$

for every compactly supported smooth function f. Thus, formally one has

$${\displaystyle \delta (x)=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{inx}}$$

on the interval [−π,π].

Despite this, the result does not hold for all compactly supported continuous functions: that is D_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

$${\displaystyle F_N(x)=\frac{1}{N}\sum _{n=0}^{N-1}{D}_n(x)=\frac{1}{N}{\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)}^2.}$$

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

$${\displaystyle {\int }_{-\pi }^{\pi }{F}_N(x)f(x)dx\to 2\pi f(0)}$$

for every compactly supported continuous 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 L², and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of L² 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 R implies that any square-integrable function f such that

$${\displaystyle \Vert f{\Vert }_{H^1}^2={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|\hat{f}(\xi ){|}^2(1+|\xi {|}^2)d\xi <\mathrm{\infty }}$$

is automatically continuous, and satisfies in particular

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

Thus δ is a bounded linear functional on the Sobolev space H¹. Equivalently δ is an element of the continuous dual space H⁻¹ of H¹. More generally, in n dimensions, one has δ ∈ H^−s(Rⁿ) provided s > n/2.

Spaces of holomorphic functions

In complex analysis, the delta function enters via Cauchy's integral formula, which asserts that if D is a domain in the complex plane with smooth boundary, then

$${\displaystyle f(z)=\frac{1}{2\pi i}{\oint }_{\mathrm{\partial }D}\frac{f(\zeta )d\zeta }{\zeta -z},z\in D}$$

for all holomorphic functions f in D that are continuous on the closure of D. As a result, the delta function δ_z is represented in this class of holomorphic functions by the Cauchy integral:

$${\displaystyle {\delta }_z[f]=f(z)=\frac{1}{2\pi i}{\oint }_{\mathrm{\partial }D}\frac{f(\zeta )d\zeta }{\zeta -z}.}$$

Moreover, let H²(∂D) be the Hardy space consisting of the closure in L²(∂D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H²(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral formula continues to hold. In particular for z ∈ D, the delta function δ_z is a continuous linear functional on H²(∂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 {φ_n} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as

$${\displaystyle f=\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{\phi }_n.}$$

The coefficients {α_n} are found as

$${\displaystyle {\alpha }_n=⟨{\phi }_n,f⟩,}$$

which may be represented by the notation:

$${\displaystyle {\alpha }_n={\phi }_n^{†}f,}$$

a form of the bra–ket notation of Dirac. Adopting this notation, the expansion of f takes the dyadic form:

$${\displaystyle f=\sum _{n=1}^{\mathrm{\infty }}{\phi }_n\left({\phi }_n^{†}f\right).}$$

Letting I denote the identity operator on the Hilbert space, the expression

$${\displaystyle I=\sum _{n=1}^{\mathrm{\infty }}{\phi }_n{\phi }_n^{†},}$$

is called a resolution of the identity. When the Hilbert space is the space L²(D) of square-integrable functions on a domain D, the quantity:

$${\displaystyle {\phi }_n{\phi }_n^{†},}$$

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

$${\displaystyle f(x)=\sum _{n=1}^{\mathrm{\infty }}{\int }_D\left({\phi }_n(x){\phi }_n^{\ast }(\xi )\right)f(\xi )d\xi .}$$

The right-hand side converges to f in the L² 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

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

resulting in the representation of the delta function:

$${\displaystyle \delta (x-\xi )=\sum _{n=1}^{\mathrm{\infty }}{\phi }_n(x){\phi }_n^{\ast }(\xi ).}$$

With a suitable rigged Hilbert space (Φ, L²(D), Φ*) where Φ ⊂ L²(D) contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φ_n. 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 α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δ_α 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 Yamashita (2007) 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

Main article: Dirac comb

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T

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,

$${\displaystyle \text{Ш}(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x-n),}$$

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

$${\displaystyle (f\ast \text{Ш})(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}f(x-n).}$$

In particular,

$${\displaystyle (f\ast \text{Ш}{)}^{\wedge }=\hat{f}\widehat{\text{Ш}}=\hat{f}\text{Ш}}$$

is precisely the Poisson summation formula. More generally, this formula remains to be true if f is a tempered distribution of rapid descent or, equivalently, if ${\displaystyle \hat{f}}$ 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 p.v. 1/x, the Cauchy principal value of the function 1/x, defined by

$${\displaystyle ⟨\mathrm{p}.\mathrm{v}.\frac{1}{x},\phi ⟩=\underset{\epsilon \to 0^{+}}{lim}{\int }_{|x|>\epsilon }\frac{\phi (x)}{x}dx.}$$

Sokhotsky's formula states that

$${\displaystyle \underset{\epsilon \to 0^{+}}{lim}\frac{1}{x\pm i\epsilon }=\mathrm{p}.\mathrm{v}.\frac{1}{x}\mp i\pi \delta (x),}$$

Here the limit is understood in the distribution sense, that for all compactly supported smooth functions f,

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{\epsilon \to 0^{+}}{lim}\frac{f(x)}{x\pm i\epsilon }dx=\mp i\pi f(0)+\underset{\epsilon \to 0^{+}}{lim}{\int }_{|x|>\epsilon }\frac{f(x)}{x}dx.}$$

Relationship to the Kronecker delta

The Kronecker delta δ_ij is the quantity defined by

$${\displaystyle {\delta }_{ij}=\{\begin{array}{ll}1& i=j\\ 0& i\ne j\end{array}}$$

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

$${\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i{\delta }_{ik}=a_k.}$$

Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\delta (x-x_0)dx=f(x_0).}$$

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

Applications

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 f(x) of a discrete distribution consisting of points x = {x₁, ..., x_n}, with corresponding probabilities p₁, ..., p_n, can be written as

$${\displaystyle f(x)=\sum _{i=1}^n{p}_i\delta (x-x_i).}$$

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

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

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 Y = g(X) is a continuous differentiable function, then the density of Y can be written as

$${\displaystyle f_Y(y)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{f}_X(x)\delta (y-g(x))dx.}$$

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 B(t) is given by

$${\displaystyle \ell (x,t)={\int }_0^t\delta (x-B(s))ds}$$

and represents the amount of time that the process spends at the point x in the range of the process. More precisely, in one dimension this integral can be written

$${\displaystyle \ell (x,t)=\underset{\epsilon \to 0^{+}}{lim}\frac{1}{2\epsilon }{\int }_0^t{\mathbf{1}}_{[x-\epsilon ,x+\epsilon ]}(B(s))ds}$$

where ${\displaystyle {\mathbf{1}}_{[x-\epsilon ,x+\epsilon ]}}$ is the indicator function of the interval ${\displaystyle [x-\epsilon ,x+\epsilon ].}$

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 L² 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 {|φ_n⟩} of wave functions is orthonormal if they are normalized by

$${\displaystyle ⟨{\phi }_n\mid {\phi }_m⟩={\delta }_{nm}}$$

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

$${\displaystyle \psi =\sum c_n{\phi }_n,}$$

with c_n = ⟨φ_n|ψ⟩. 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, as above, this equality implies the resolution of the identity:

$${\displaystyle I=\sum |{\phi }_n⟩⟨{\phi }_n|.}$$

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an observable may be continuous rather than discrete. An example is the position observable, Qψ(x) = xψ(x). 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: that is, to replace the Hilbert space of quantum mechanics with an appropriate rigged Hilbert space. In this context, the position operator has a complete set of eigen-distributions, labeled by the points y of the real line, given by

$${\displaystyle {\phi }_y(x)=\delta (x-y).}$$

The eigenfunctions of position are denoted by φ_y = |y⟩ in Dirac notation, and are known as position eigenstates.

Similar considerations apply to the eigenstates of the momentum operator, or indeed any other self-adjoint unbounded operator P on the Hilbert space, provided the spectrum of P is continuous and there are no degenerate eigenvalues. In that case, there is a set Ω of real numbers (the spectrum), and a collection φ_y of distributions indexed by the elements of Ω, such that

$${\displaystyle P{\phi }_y=y{\phi }_y.}$$

That is, φ_y are the eigenvectors of P. If the eigenvectors are normalized so that

$${\displaystyle ⟨{\phi }_y,{\phi }_{y^{\prime }}⟩=\delta (y-y^{\prime })}$$

in the distribution sense, then for any test function ψ,

$${\displaystyle \psi (x)={\int }_{\mathrm{\Omega }}c(y){\phi }_y(x)dy}$$

where c(y) = ⟨ψ, φ_y⟩. That is, as in the discrete case, there is a resolution of the identity

$${\displaystyle I={\int }_{\mathrm{\Omega }}|{\phi }_y⟩⟨{\phi }_y|dy}$$

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 I at time t = 0 can be written

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

where m is the mass, ξ is the deflection, and k is the spring constant.

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

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

where EI is the bending stiffness of the beam, w is the deflection, x is the spatial coordinate, and q(x) is the load distribution. If a beam is loaded by a point force F at x = x₀, the load distribution is written

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

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 F at a distance d apart. They then produce a moment M = Fd acting on the beam. Now, let the distance d approach the limit zero, while M is kept constant. The load distribution, assuming a clockwise moment acting at x = 0, is written

$${\displaystyle \begin{array}{rl}q(x)& =\underset{d\to 0}{lim}(F\delta (x)-F\delta (x-d))\\ & =\underset{d\to 0}{lim}\left(\frac{M}{d}\delta (x)-\frac{M}{d}\delta (x-d)\right)\\ & =M\underset{d\to 0}{lim}\frac{\delta (x)-\delta (x-d)}{d}\\ & =M{\delta }^{\prime }(x).\end{array}}$$

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

Coordination complex

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Coordination_complex

Cisplatin, PtCl₂(NH₃)₂, is a coordination complex of platinum(II) with two chloride and two ammonia ligands. It is one of the most successful anticancer drugs.

A coordination complex is a chemical compound consisting of a central atom or ion, which is usually metallic and is called the coordination centre, and a surrounding array of bound molecules or ions, that are in turn known as ligands or complexing agents. Many metal-containing compounds, especially those that include transition metals (elements like titanium that belong to the periodic table's d-block), are coordination complexes.

Nomenclature and terminology

Coordination complexes are so pervasive that their structures and reactions are described in many ways, sometimes confusingly. The atom within a ligand that is bonded to the central metal atom or ion is called the donor atom. In a typical complex, a metal ion is bonded to several donor atoms, which can be the same or different. A polydentate (multiple bonded) ligand is a molecule or ion that bonds to the central atom through several of the ligand's atoms; ligands with 2, 3, 4 or even 6 bonds to the central atom are common. These complexes are called chelate complexes; the formation of such complexes is called chelation, complexation, and coordination.

The central atom or ion, together with all ligands, comprise the coordination sphere. The central atoms or ion and the donor atoms comprise the first coordination sphere.

Coordination refers to the "coordinate covalent bonds" (dipolar bonds) between the ligands and the central atom. Originally, a complex implied a reversible association of molecules, atoms, or ions through such weak chemical bonds. As applied to coordination chemistry, this meaning has evolved. Some metal complexes are formed virtually irreversibly and many are bound together by bonds that are quite strong.

The number of donor atoms attached to the central atom or ion is called the coordination number. The most common coordination numbers are 2, 4, and especially 6. A hydrated ion is one kind of a complex ion (or simply a complex), a species formed between a central metal ion and one or more surrounding ligands, molecules or ions that contain at least one lone pair of electrons.

If all the ligands are monodentate, then the number of donor atoms equals the number of ligands. For example, the cobalt(II) hexahydrate ion or the hexaaquacobalt(II) ion [Co(H₂O)₆]²⁺ is a hydrated-complex ion that consists of six water molecules attached to a metal ion Co. The oxidation state and the coordination number reflect the number of bonds formed between the metal ion and the ligands in the complex ion. However, the coordination number of Pt(en)²⁺
₂ is 4 (rather than 2) since it has two bidentate ligands, which contain four donor atoms in total.

Any donor atom will give a pair of electrons. There are some donor atoms or groups which can offer more than one pair of electrons. Such are called bidentate (offers two pairs of electrons) or polydentate (offers more than two pairs of electrons). In some cases an atom or a group offers a pair of electrons to two similar or different central metal atoms or acceptors—by division of the electron pair—into a three-center two-electron bond. These are called bridging ligands.

History

Alfred Werner

Coordination complexes have been known since the beginning of modern chemistry. Early well-known coordination complexes include dyes such as Prussian blue. Their properties were first well understood in the late 1800s, following the 1869 work of Christian Wilhelm Blomstrand. Blomstrand developed what has come to be known as the complex ion chain theory. In considering metal amine complexes, he theorized that the ammonia molecules compensated for the charge of the ion by forming chains of the type [(NH₃)_X]^X+, where X is the coordination number of the metal ion. He compared his theoretical ammonia chains to hydrocarbons of the form (CH₂)_X.

Following this theory, Danish scientist Sophus Mads Jørgensen made improvements to it. In his version of the theory, Jørgensen claimed that when a molecule dissociates in a solution there were two possible outcomes: the ions would bind via the ammonia chains Blomstrand had described or the ions would bind directly to the metal.

It was not until 1893 that the most widely accepted version of the theory today was published by Alfred Werner. Werner's work included two important changes to the Blomstrand theory. The first was that Werner described the two possibilities in terms of location in the coordination sphere. He claimed that if the ions were to form a chain, this would occur outside of the coordination sphere while the ions that bound directly to the metal would do so within the coordination sphere. In one of his most important discoveries however Werner disproved the majority of the chain theory. Werner discovered the spatial arrangements of the ligands that were involved in the formation of the complex hexacoordinate cobalt. His theory allows one to understand the difference between a coordinated ligand and a charge balancing ion in a compound, for example the chloride ion in the cobaltammine chlorides and to explain many of the previously inexplicable isomers.

In 1911, Werner first resolved the coordination complex hexol into optical isomers, overthrowing the theory that only carbon compounds could possess chirality.

Structures

Structure of hexol

The ions or molecules surrounding the central atom are called ligands. Ligands are classified as L or X (or a combination thereof), depending on how many electrons they provide for the bond between ligand and central atom. L ligands provide two electrons from a lone electron pair, resulting in a coordinate covalent bond. X ligands provide one electron, with the central atom providing the other electron, thus forming a regular covalent bond. The ligands are said to be coordinated to the atom. For alkenes, the pi bonds can coordinate to metal atoms. An example is ethylene in the complex [PtCl₃(C₂H₄)]⁻ (Zeise's salt).

Geometry

In coordination chemistry, a structure is first described by its coordination number, the number of ligands attached to the metal (more specifically, the number of donor atoms). Usually one can count the ligands attached, but sometimes even the counting can become ambiguous. Coordination numbers are normally between two and nine, but large numbers of ligands are not uncommon for the lanthanides and actinides. The number of bonds depends on the size, charge, and electron configuration of the metal ion and the ligands. Metal ions may have more than one coordination number.

Typically the chemistry of transition metal complexes is dominated by interactions between s and p molecular orbitals of the donor-atoms in the ligands and the d orbitals of the metal ions. The s, p, and d orbitals of the metal can accommodate 18 electrons (see 18-Electron rule). The maximum coordination number for a certain metal is thus related to the electronic configuration of the metal ion (to be more specific, the number of empty orbitals) and to the ratio of the size of the ligands and the metal ion. Large metals and small ligands lead to high coordination numbers, e.g. [Mo(CN)₈]⁴⁻. Small metals with large ligands lead to low coordination numbers, e.g. Pt[P(CMe₃)]₂. Due to their large size, lanthanides, actinides, and early transition metals tend to have high coordination numbers.

Most structures follow the points-on-a-sphere pattern (or, as if the central atom were in the middle of a polyhedron where the corners of that shape are the locations of the ligands), where orbital overlap (between ligand and metal orbitals) and ligand-ligand repulsions tend to lead to certain regular geometries. The most observed geometries are listed below, but there are many cases that deviate from a regular geometry, e.g. due to the use of ligands of diverse types (which results in irregular bond lengths; the coordination atoms do not follow a points-on-a-sphere pattern), due to the size of ligands, or due to electronic effects (see, e.g., Jahn–Teller distortion):

• Linear for two-coordination
• Trigonal planar for three-coordination
• Tetrahedral or square planar for four-coordination
• Trigonal bipyramidal for five-coordination
• Octahedral for six-coordination
• Pentagonal bipyramidal for seven-coordination
• Square antiprismatic for eight-coordination
• Tricapped trigonal prismatic for nine-coordination

The idealized descriptions of 5-, 7-, 8-, and 9- coordination are often indistinct geometrically from alternative structures with slightly differing L-M-L (ligand-metal-ligand) angles, e.g. the difference between square pyramidal and trigonal bipyramidal structures.

• Square pyramidal for five-coordination
• Capped octahedral or capped trigonal prismatic for seven-coordination
• Dodecahedral or bicapped trigonal prismatic for eight-coordination
• Capped square antiprismatic for nine-coordination

To distinguish between the alternative coordinations for five-coordinated complexes, the τ geometry index was invented by Addison et al. This index depends on angles by the coordination center and changes between 0 for the square pyramidal to 1 for trigonal bipyramidal structures, allowing to classify the cases in between. This system was later extended to four-coordinated complexes by Houser et al. and also Okuniewski et al.

In systems with low d electron count, due to special electronic effects such as (second-order) Jahn–Teller stabilization, certain geometries (in which the coordination atoms do not follow a points-on-a-sphere pattern) are stabilized relative to the other possibilities, e.g. for some compounds the trigonal prismatic geometry is stabilized relative to octahedral structures for six-coordination.

• Bent for two-coordination
• Trigonal pyramidal for three-coordination
• Trigonal prismatic for six-coordination

Isomerism

The arrangement of the ligands is fixed for a given complex, but in some cases it is mutable by a reaction that forms another stable isomer.

There exist many kinds of isomerism in coordination complexes, just as in many other compounds.

Stereoisomerism

Stereoisomerism occurs with the same bonds in distinct orientations. Stereoisomerism can be further classified into:

Cis–trans isomerism and facial–meridional isomerism

Cis–trans isomerism occurs in octahedral and square planar complexes (but not tetrahedral). When two ligands are adjacent they are said to be cis, when opposite each other, trans. When three identical ligands occupy one face of an octahedron, the isomer is said to be facial, or fac. In a fac isomer, any two identical ligands are adjacent or cis to each other. If these three ligands and the metal ion are in one plane, the isomer is said to be meridional, or mer. A mer isomer can be considered as a combination of a trans and a cis, since it contains both trans and cis pairs of identical ligands.

• 

  cis-[CoCl₂(NH₃)₄]⁺

• 

  trans-[CoCl₂(NH₃)₄]⁺

• 

  fac-[CoCl₃(NH₃)₃]

• 

  mer-[CoCl₃(NH₃)₃]

Optical isomerism

Optical isomerism occurs when a complex is not superimposable with its mirror image. It is so called because the two isomers are each optically active, that is, they rotate the plane of polarized light in opposite directions. In the first molecule shown, the symbol Λ (lambda) is used as a prefix to describe the left-handed propeller twist formed by three bidentate ligands. The second molecule is the mirror image of the first, with the symbol Δ (delta) as a prefix for the right-handed propeller twist. The third and fourth molecules are a similar pair of Λ and Δ isomers, in this case with two bidentate ligands and two identical monodentate ligands.

• 

  Λ-[Fe(ox)₃]³⁻

• 

  Δ-[Fe(ox)₃]³⁻

• 

  Λ-cis-[CoCl₂(en)₂]⁺

• 

  Δ-cis-[CoCl₂(en)₂]⁺

Structural isomerism

Structural isomerism occurs when the bonds are themselves different. Four types of structural isomerism are recognized: ionisation isomerism, solvate or hydrate isomerism, linkage isomerism and coordination isomerism.

1. Ionisation isomerism – the isomers give different ions in solution although they have the same composition. This type of isomerism occurs when the counter ion of the complex is also a potential ligand. For example, pentaamminebromocobalt(III) sulphate [Co(NH₃)₅Br]SO₄ is red violet and in solution gives a precipitate with barium chloride, confirming the presence of sulphate ion, while pentaamminesulphatecobalt(III) bromide [Co(NH₃)₅SO₄]Br is red and tests negative for sulphate ion in solution, but instead gives a precipitate of AgBr with silver nitrate.
2. Solvate or hydrate isomerism – the isomers have the same composition but differ with respect to the number of molecules of solvent that serve as ligand vs simply occupying sites in the crystal. Examples: [Cr(H₂O)₆]Cl₃ is violet colored, [CrCl(H₂O)₅]Cl₂·H₂O is blue-green, and [CrCl₂(H₂O)₄]Cl·2H₂O is dark green. See water of crystallization.
3. Linkage isomerism occurs with ligands with more than one possible donor atom, known as ambidentate ligands. For example, nitrite can coordinate through O or N. One pair of nitrite linkage isomers have structures (NH₃)₅CoNO2+2 (nitro isomer) and (NH₃)₅CoONO²⁺ (nitrito isomer).
4. Coordination isomerism – this occurs when both positive and negative ions of a salt are complex ions and the two isomers differ in the distribution of ligands between the cation and the anion. For example, [Co(NH₃)₆][Cr(CN)₆] and [Cr(NH₃)₆][Co(CN)₆].

Electronic properties

Many of the properties of transition metal complexes are dictated by their electronic structures. The electronic structure can be described by a relatively ionic model that ascribes formal charges to the metals and ligands. This approach is the essence of crystal field theory (CFT). Crystal field theory, introduced by Hans Bethe in 1929, gives a quantum mechanically based attempt at understanding complexes. But crystal field theory treats all interactions in a complex as ionic and assumes that the ligands can be approximated by negative point charges.

More sophisticated models embrace covalency, and this approach is described by ligand field theory (LFT) and Molecular orbital theory (MO). Ligand field theory, introduced in 1935 and built from molecular orbital theory, can handle a broader range of complexes and can explain complexes in which the interactions are covalent. The chemical applications of group theory can aid in the understanding of crystal or ligand field theory, by allowing simple, symmetry based solutions to the formal equations.

Chemists tend to employ the simplest model required to predict the properties of interest; for this reason, CFT has been a favorite for the discussions when possible. MO and LF theories are more complicated, but provide a more realistic perspective.

The electronic configuration of the complexes gives them some important properties:

Synthesis of copper(II)-tetraphenylporphyrin, a metal complex, from tetraphenylporphyrin and copper(II) acetate monohydrate.

Color of transition metal complexes

Transition metal complexes often have spectacular colors caused by electronic transitions by the absorption of light. For this reason they are often applied as pigments. Most transitions that are related to colored metal complexes are either d–d transitions or charge transfer bands. In a d–d transition, an electron in a d orbital on the metal is excited by a photon to another d orbital of higher energy, therefore d–d transitions occur only for partially-filled d-orbital complexes (d^1–9). For complexes having d⁰ or d¹⁰ configuration, charge transfer is still possible even though d–d transitions are not. A charge transfer band entails promotion of an electron from a metal-based orbital into an empty ligand-based orbital (metal-to-ligand charge transfer or MLCT). The converse also occurs: excitation of an electron in a ligand-based orbital into an empty metal-based orbital (ligand-to-metal charge transfer or LMCT). These phenomena can be observed with the aid of electronic spectroscopy; also known as UV-Vis. For simple compounds with high symmetry, the d–d transitions can be assigned using Tanabe–Sugano diagrams. These assignments are gaining increased support with computational chemistry.

Colours of Various Example Coordination Complexes

	Fe2+	Fe3+	Co2+	Cu2+	Al3+	Cr3+
Hydrated Ion	[Fe(H2O)6]2+ Pale green Solution	[Fe(H2O)6]3+ Yellow/brown Solution	[Co(H2O)6]2+ Pink Solution	[Cu(H2O)6]2+ Blue Solution	[Al(H2O)6]3+ Colourless Solution	[Cr(H2O)6]3+ Green Solution
(OH)−, dilute	[Fe(H2O)4(OH)2] Dark green Precipitate	[Fe(H2O)3(OH)3] Brown Precipitate	[Co(H2O)4(OH)2] Blue/green Precipitate	[Cu(H2O)4(OH)2] Blue Precipitate	[Al(H2O)3(OH)3] White Precipitate	[Cr(H2O)3(OH)3] Green Precipitate
(OH)−, concentrated	[Fe(H2O)4(OH)2] Dark green Precipitate	[Fe(H2O)3(OH)3] Brown Precipitate	[Co(H2O)4(OH)2] Blue/green Precipitate	[Cu(H2O)4(OH)2] Blue Precipitate	[Al(OH)4]− Colourless Solution	[Cr(OH)6]3− Green Solution
NH3, dilute	[Fe(NH3)6]2+ Dark green Precipitate	[Fe(NH3)6]3+ Brown Precipitate	[Co(NH3)6]2+ Straw coloured Solution	[Cu(NH3)4(H2O)2]2+ Deep blue Solution	[Al(NH3)3]3+ White Precipitate	[Cr(NH3)6]3+ Purple Solution
NH3, concentrated	[Fe(NH3)6]2+ Dark green Precipitate	[Fe(NH3)6]3+ Brown Precipitate	[Co(NH3)6]2+ Straw coloured Solution	[Cu(NH3)4(H2O)2]2+ Deep blue Solution	[Al(NH3)3]3+ White Precipitate	[Cr(NH3)6]3+ Purple Solution
(CO3)2-	FeCO3 Dark green Precipitate	Fe2(CO3)3 Brown Precipitate+bubbles	CoCO3 Pink Precipitate	CuCO3 Blue/green Precipitate

Colors of lanthanide complexes

Superficially lanthanide complexes are similar to those of the transition metals in that some are colored. However, for the common Ln³⁺ ions (Ln = lanthanide) the colors are all pale, and hardly influenced by the nature of the ligand. The colors are due to 4f electron transitions. As the 4f orbitals in lanthanides are "buried" in the xenon core and shielded from the ligand by the 5s and 5p orbitals they are therefore not influenced by the ligands to any great extent leading to a much smaller crystal field splitting than in the transition metals. The absorption spectra of an Ln³⁺ ion approximates to that of the free ion where the electronic states are described by spin-orbit coupling. This contrasts to the transition metals where the ground state is split by the crystal field. Absorptions for Ln³⁺ are weak as electric dipole transitions are parity forbidden (Laporte forbidden) but can gain intensity due to the effect of a low-symmetry ligand field or mixing with higher electronic states (e.g. d orbitals). f-f absorption bands are extremely sharp which contrasts with those observed for transition metals which generally have broad bands. This can lead to extremely unusual effects, such as significant color changes under different forms of lighting.

Magnetism

Main article: magnetochemistry

Metal complexes that have unpaired electrons are magnetic. Considering only monometallic complexes, unpaired electrons arise because the complex has an odd number of electrons or because electron pairing is destabilized. Thus, monomeric Ti(III) species have one "d-electron" and must be (para)magnetic, regardless of the geometry or the nature of the ligands. Ti(II), with two d-electrons, forms some complexes that have two unpaired electrons and others with none. This effect is illustrated by the compounds TiX₂[(CH₃)₂PCH₂CH₂P(CH₃)₂]₂: when X = Cl, the complex is paramagnetic (high-spin configuration), whereas when X = CH₃, it is diamagnetic (low-spin configuration). It is important to realize that ligands provide an important means of adjusting the ground state properties.

In bi- and polymetallic complexes, in which the individual centres have an odd number of electrons or that are high-spin, the situation is more complicated. If there is interaction (either direct or through ligand) between the two (or more) metal centres, the electrons may couple (antiferromagnetic coupling, resulting in a diamagnetic compound), or they may enhance each other (ferromagnetic coupling). When there is no interaction, the two (or more) individual metal centers behave as if in two separate molecules.

Reactivity

Complexes show a variety of possible reactivities:

• Electron transfers

  Electron transfer (ET) between metal ions can occur via two distinct mechanisms, inner and outer sphere electron transfers. In an inner sphere reaction, a bridging ligand serves as a conduit for ET.

• (Degenerate) ligand exchange

  One important indicator of reactivity is the rate of degenerate exchange of ligands. For example, the rate of interchange of coordinate water in [M(H₂O)₆]ⁿ⁺ complexes varies over 20 orders of magnitude. Complexes where the ligands are released and rebound rapidly are classified as labile. Such labile complexes can be quite stable thermodynamically. Typical labile metal complexes either have low-charge (Na⁺), electrons in d-orbitals that are antibonding with respect to the ligands (Zn²⁺), or lack covalency (Ln³⁺, where Ln is any lanthanide). The lability of a metal complex also depends on the high-spin vs. low-spin configurations when such is possible. Thus, high-spin Fe(II) and Co(III) form labile complexes, whereas low-spin analogues are inert. Cr(III) can exist only in the low-spin state (quartet), which is inert because of its high formal oxidation state, absence of electrons in orbitals that are M–L antibonding, plus some "ligand field stabilization" associated with the d³ configuration.

• Associative processes

  Complexes that have unfilled or half-filled orbitals are often capable of reacting with substrates. Most substrates have a singlet ground-state; that is, they have lone electron pairs (e.g., water, amines, ethers), so these substrates need an empty orbital to be able to react with a metal centre. Some substrates (e.g., molecular oxygen) have a triplet ground state, which results that metals with half-filled orbitals have a tendency to react with such substrates (it must be said that the dioxygen molecule also has lone pairs, so it is also capable to react as a 'normal' Lewis base).

If the ligands around the metal are carefully chosen, the metal can aid in (stoichiometric or catalytic) transformations of molecules or be used as a sensor.

Classification

Metal complexes, also known as coordination compounds, include virtually all metal compounds. The study of "coordination chemistry" is the study of "inorganic chemistry" of all alkali and alkaline earth metals, transition metals, lanthanides, actinides, and metalloids. Thus, coordination chemistry is the chemistry of the majority of the periodic table. Metals and metal ions exist, in the condensed phases at least, only surrounded by ligands.

The areas of coordination chemistry can be classified according to the nature of the ligands, in broad terms:

• Classical (or "Werner Complexes"): Ligands in classical coordination chemistry bind to metals, almost exclusively, via their lone pairs of electrons residing on the main-group atoms of the ligand. Typical ligands are H₂O, NH₃, Cl⁻, CN⁻, en. Some of the simplest members of such complexes are described in metal aquo complexes, metal ammine complexes,

Examples: [Co(EDTA)]⁻, [Co(NH₃)₆]³⁺, [Fe(C₂O₄)₃]^3-

• Organometallic chemistry: Ligands are organic (alkenes, alkynes, alkyls) as well as "organic-like" ligands such as phosphines, hydride, and CO.

Example: (C₅H₅)Fe(CO)₂CH₃

• Bioinorganic chemistry: Ligands are those provided by nature, especially including the side chains of amino acids, and many cofactors such as porphyrins.

Example: hemoglobin contains heme, a porphyrin complex of iron

Example: chlorophyll contains a porphyrin complex of magnesium

Many natural ligands are "classical" especially including water.

• Cluster chemistry: Ligands include all of the above as well as other metal ions or atoms as well.

Example Ru₃(CO)₁₂

• In some cases there are combinations of different fields:

Example: [Fe₄S₄(Scysteinyl)₄]²⁻, in which a cluster is embedded in a biologically active species.

Mineralogy, materials science, and solid state chemistry – as they apply to metal ions – are subsets of coordination chemistry in the sense that the metals are surrounded by ligands. In many cases these ligands are oxides or sulfides, but the metals are coordinated nonetheless, and the principles and guidelines discussed below apply. In hydrates, at least some of the ligands are water molecules. It is true that the focus of mineralogy, materials science, and solid state chemistry differs from the usual focus of coordination or inorganic chemistry. The former are concerned primarily with polymeric structures, properties arising from a collective effects of many highly interconnected metals. In contrast, coordination chemistry focuses on reactivity and properties of complexes containing individual metal atoms or small ensembles of metal atoms.

Nomenclature of coordination complexes

The basic procedure for naming a complex is:

1. When naming a complex ion, the ligands are named before the metal ion.
2. The ligands' names are given in alphabetical order. Numerical prefixes do not affect the order.
   • Multiple occurring monodentate ligands receive a prefix according to the number of occurrences: di-, tri-, tetra-, penta-, or hexa-.
   • Multiple occurring polydentate ligands (e.g., ethylenediamine, oxalate) receive bis-, tris-, tetrakis-, etc.
   • Anions end in o. This replaces the final 'e' when the anion ends with '-ide', '-ate' or '-ite', e.g. chloride becomes chlorido and sulfate becomes sulfato. Formerly, '-ide' was changed to '-o' (e.g. chloro and cyano), but this rule has been modified in the 2005 IUPAC recommendations and the correct forms for these ligands are now chlorido and cyanido.
   • Neutral ligands are given their usual name, with some exceptions: NH₃ becomes ammine; H₂O becomes aqua or aquo; CO becomes carbonyl; NO becomes nitrosyl.
3. Write the name of the central atom/ion. If the complex is an anion, the central atom's name will end in -ate, and its Latin name will be used if available (except for mercury).
4. The oxidation state of the central atom is to be specified (when it is one of several possible, or zero), and should be written as a Roman numeral (or 0) enclosed in parentheses.
5. Name of the cation should be preceded by the name of anion. (if applicable, as in last example)

Examples:

metal	changed to
cobalt	cobaltate
aluminium	aluminate
chromium	chromate
vanadium	vanadate
copper	cuprate
iron	ferrate

[Cd(CN)₂(en)₂] → dicyanidobis(ethylenediamine)cadmium(II)

[CoCl(NH₃)₅]SO₄ → pentaamminechloridocobalt(III) sulfate

[Cu(H₂O)₆] ²⁺ → hexaaquacopper(II) ion

[CuCl₅NH₃]³⁻ → amminepentachloridocuprate(II) ion

K₄[Fe(CN)₆] → potassium hexacyanidoferrate(II)

[NiCl₄]²⁻ → tetrachloridonickelate(II) ion (The use of chloro- was removed from IUPAC naming convention)

The coordination number of ligands attached to more than one metal (bridging ligands) is indicated by a subscript to the Greek symbol μ placed before the ligand name. Thus the dimer of aluminium trichloride is described by Al₂Cl₄(μ₂-Cl)₂.

Any anionic group can be electronically stabilized by any cation. An anionic complex can be stabilised by a hydrogen cation, becoming an acidic complex which can dissociate to release the cationic hydrogen. This kind of complex compound has a name with "ic" added after the central metal. For example, H₂[Pt(CN)₄] has the name tetracyanoplatinic (II) acid.

Stability constant

Main article: Stability constants of complexes

The affinity of metal ions for ligands is described by a stability constant, also called the formation constant, and is represented by the symbol K_f. It is the equilibrium constant for its assembly from the constituent metal and ligands, and can be calculated accordingly, as in the following example for a simple case:

xM _(aq) + yL _(aq) ⇌ zZ _(aq)

${\displaystyle K_f=\frac{[\text{Z}{]}^z}{[\text{M}{]}^x[\text{L}{]}^y}}$

where : x, y, and z are the stoichiometric coefficients of each species. M stands for metal / metal ion , the L for Lewis bases , and finally Z for complex ions. Formation constants vary widely. Large values indicate that the metal has high affinity for the ligand, provided the system is at equilibrium.

Sometimes the stability constant will be in a different form known as the constant of destability. This constant is expressed as the inverse of the constant of formation and is denoted as K_d = 1/K_f . This constant represents the reverse reaction for the decomposition of a complex ion into its individual metal and ligand components. When comparing the values for K_d, the larger the value, the more unstable the complex ion is.

As a result of these complex ions forming in solutions they also can play a key role in solubility of other compounds. When a complex ion is formed it can alter the concentrations of its components in the solution. For example:

Ag⁺
_(aq) + 2 NH₃ ⇌ Ag(NH₃)⁺
₂

AgCl_(s) + H₂O_(l) ⇌ Ag⁺
_(aq) + Cl⁻
_(aq)

If these reactions both occurred in the same reaction vessel, the solubility of the silver chloride would be increased by the presence of NH₄OH because formation of the Diammine argentum(I) complex consumes a significant portion of the free silver ions from the solution. By Le Chatelier's principle, this causes the equilibrium reaction for the dissolving of the silver chloride, which has silver ion as a product, to shift to the right.

This new solubility can be calculated given the values of K_f and K_sp for the original reactions. The solubility is found essentially by combining the two separate equilibria into one combined equilibrium reaction and this combined reaction is the one that determines the new solubility. So K_c, the new solubility constant, is denoted by:

${\displaystyle K_c=K_{sp}{K}_f}$

Application of coordination compounds

As metals only exist in solution as coordination complexes, it follows then that this class of compounds is useful in a wide variety of ways.

Bioinorganic chemistry

In bioinorganic chemistry and bioorganometallic chemistry, coordination complexes serve either structural or catalytic functions. An estimated 30% of proteins contain metal ions. Examples include the intensely colored vitamin B₁₂, the heme group in hemoglobin, the cytochromes, the chlorin group in chlorophyll, and carboxypeptidase, a hydrolytic enzyme important in digestion. Another complex ion enzyme is catalase, which decomposes the cell's waste hydrogen peroxide. Synthetic coordination compounds are also used to bind to proteins and especially nucleic acids (e.g. anticancer drug cisplatin).

Industry

Homogeneous catalysis is a major application of coordination compounds for the production of organic substances. Processes include hydrogenation, hydroformylation, oxidation. In one example, a combination of titanium trichloride and triethylaluminium gives rise to Ziegler–Natta catalysts, used for the polymerization of ethylene and propylene to give polymers of great commercial importance as fibers, films, and plastics.

Nickel, cobalt, and copper can be extracted using hydrometallurgical processes involving complex ions. They are extracted from their ores as ammine complexes. Metals can also be separated using the selective precipitation and solubility of complex ions. Cyanide is used chiefly for extraction of gold and silver from their ores.

Phthalocyanine complexes are an important class of pigments.

Analysis

At one time, coordination compounds were used to identify the presence of metals in a sample. Qualitative inorganic analysis has largely been superseded by instrumental methods of analysis such as atomic absorption spectroscopy (AAS), inductively coupled plasma atomic emission spectroscopy (ICP-AES) and inductively coupled plasma mass spectrometry (ICP-MS).