Is not a function, but a distribution (functional analysis). In the context of distribution (functional analysis) it is defined as:
$$ <\delta,\varphi>=\varphi(0) $$Informally, Dirac delta function is usually written as
$$ \delta(x)=\left\{\begin{array}{ll}{+\infty,} & {x=0} \\ {0,} & {x \neq 0}\end{array}\right. $$More on the Dirac delta: it picks out the value of $\varphi$ at 0 ignoring the others, so in some sense it is a continuous version of the Kronecker delta! That is, in the same sense that
$$ \delta_{ij}=\left\{\begin{array}{ll}{1,} & {i=j} \\ {0,} & {i \neq j}\end{array}\right. $$we have that
$$ \delta_y(x):=\delta(x-y)=\left\{\begin{array}{ll}{+\infty,} & {x=y} \\ {0,} & {x \neq y}\end{array}\right. $$Even more on this, consider for simplicity $D(\mathbb{R})$ (see distribution (functional analysis)). For every $y\in \mathbb{R}$ we can consider
$$ T_y:D(\mathbb{R}) \mapsto \mathbb{R} $$such that $T_y(\varphi)=\varphi(q)$. It can be checked that informally:
$$ T_y=\delta_y(x) $$and
$$ T_y(\varphi)=\varphi(y)=<\delta_y,\varphi>=\int_{\mathbb{R}}\varphi(x) \delta_y(x)dx. $$In this way, $\delta_y$ plays the role of $e_j$ when $\{e_i\}$ is an orthogonal basis in a vector space $V$, since for a vector $v\in V$ its $j$th coordinate is
$$ coord_j(v)=v_j=In the same sense that $\delta_{ij}$, with $i$ running from 1 to $n$, are the coordinates of $e_j$ in the basis $\{e_i\}$, the expression $\delta_y(x)$ represents the "coordinates" of $\delta_y$ in the basis $\{\delta_x\}$. The notation is not appropriate, and indeed this last sentence should be:
>the expression $\delta_y(x)$ represents the "coordinates" of $|y\rangle$ in the basis $\{\delta_x\}$
in concordance with the bra-ket notation.
The Fourier transform of the Dirac delta is
$$ \delta_a(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i w(t-a)} d w= $$ $$ =\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-iaw}e^{i wt} d w $$See Wikipedia for a proof.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: