Editing Dirac delta function (section)

====Spaces of holomorphic functions====
In [[complex analysis]], the delta function enters via [[Cauchy's integral formula]], which asserts that if {{mvar|D}} is a domain in the [[complex plane]] with smooth boundary, then

<math display="block">f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D</math>

for all [[holomorphic function]]s {{mvar|f}} in {{mvar|D}} that are continuous on the closure of {{mvar|D}}.  As a result, the delta function {{math|''δ''<sub>''z''</sub>}} is represented in this class of holomorphic functions by the Cauchy integral:

<math display="block">\delta_z[f] = f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z}.</math>

Moreover, let {{math|''H''<sup>2</sup>(∂''D'')}} be the [[Hardy space]] consisting of the closure in {{math|''L''<sup>2</sup>(∂''D'')}} of all holomorphic functions in {{mvar|D}} continuous up to the boundary of {{mvar|D}}.  Then functions in {{math|''H''<sup>2</sup>(∂''D'')}} uniquely extend to holomorphic functions in {{mvar|D}}, and the Cauchy integral formula continues to hold.  In particular for {{math|''z'' ∈ ''D''}}, the delta function {{mvar|δ<sub>z</sub>}} is a continuous linear functional on {{math|''H''<sup>2</sup>(∂''D'')}}.  This is a special case of the situation in [[several complex variables]] in which, for smooth domains {{mvar|D}}, the [[Szegő kernel]] plays the role of the Cauchy integral.{{sfn|Hazewinkel|1995|p=[{{google books |plainurl=y |id=PE1a-EIG22kC|page=357}} 357]}}

Another representation of the delta function in a space of holomorphic functions is on the space <math>H(D)\cap L^2(D)</math> of square-integrable holomorphic functions in an open set <math>D\subset\mathbb C^n</math>.  This is a closed subspace of <math>L^2(D)</math>, and therefore is a Hilbert space.  On the other hand, the functional that evaluates a holomorphic function in <math>H(D)\cap L^2(D)</math> at a point <math>z</math> of <math>D</math> is a continuous functional, and so by the Riesz representation theorem, is represented by integration against a kernel <math>K_z(\zeta)</math>, the [[Bergman kernel]].  This kernel is the analog of the delta function in this Hilbert space.  A Hilbert space having such a kernel is called a [[reproducing kernel Hilbert space]].  In the special case of the unit disc, one has
<math display="block">\delta_w[f] = f(w) = \frac1\pi\iint_{|z|<1} \frac{f(z)\,dx\,dy}{(1-\bar zw)^2}.</math>