How do I calculate the EM fields of a massless charged particle? I am trying to calculate the Electromagnetic fields from a massless charged particle. I am trying to reproduce the starting results given in the "Electromagnetic fields of a massless particle and the eikonal". I am having difficulty in calculating the following limit.
$$\lim_{v\rightarrow 1}\dfrac{1-v^2}{[(z-vt)^2 + (1-v^2)r^2]^{3/2}} = \dfrac{2}{r^2}\delta(t-z)$$
How do I calculate this? It says it could be done by taking the Fourier transform with respect to $z$ but I am not able to do so.
 A: In the referenced paper we read

It is straightforward to establish (for example by
Fourier transforming with respect to $z$) that
\begin{equation}
\boxed{\:\:
\lim_{\upsilon\boldsymbol{\rightarrow} 1}\dfrac{1\boldsymbol{-}\upsilon^2}{\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}\boldsymbol{=}\dfrac{2}{r^2_{\boldsymbol{\perp}}}\,\delta\left(t\boldsymbol{-}z\right)\:\:\vphantom{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}} 
\tag{4 of paper}\label{4 of paper}   
\end{equation}

Instead of Fourier transforming I'll try to follow a different way.
So, consider a real function $\;f(z)\;$ of the real variable $\;z\in\mathbb{R}\;$ for which
\begin{align}
f(z)\boldsymbol{=}0 \quad & \text{for any} \quad z\boldsymbol{\ne} z_{0} \quad \textbf{and}
\tag{01a}\label{01a}\\
\int\limits_{\boldsymbol{z_{0}-\varepsilon}}^{\boldsymbol{z_{0}+\varepsilon}}\!\!\!f(z)\mathrm dz\boldsymbol{=}1\quad &  \text{for any} \quad \boldsymbol{\varepsilon} \boldsymbol{>}0
\tag{01b}\label{01b} 
\end{align}
Under these conditions it seems that this function is not well-defined at  $\;z_{0}$, may be because of a singularity at this point. But we have good reasons to $^{\prime\prime}$believe$^{\prime\prime}$ that
\begin{equation}
f(z)\boldsymbol{\equiv}\delta\left(z\boldsymbol{-}z_{0}\right)
\tag{02}\label{02}
\end{equation}
since equations \eqref{01a},\eqref{01b} remind us the defining properties of Dirac delta function on the real axis $\;\mathbb{R}$.
In our case we have the function
\begin{equation}
f(z)\boldsymbol{=}\lim_{\upsilon\boldsymbol{\rightarrow} 1}\dfrac{1\boldsymbol{-}\upsilon^2}{\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}
\tag{03}\label{03}
\end{equation}
This function fulfills the condition \eqref{01a} with $z_{0}=t$
\begin{equation}
f(z)\boldsymbol{=}\lim_{\upsilon\boldsymbol{\rightarrow} 1}\dfrac{1\boldsymbol{-}\upsilon^2}{\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}\boldsymbol{=}0 \qquad\text{for any} \quad z\boldsymbol{\ne} t
\tag{04}\label{04}
\end{equation}
But this function is not well-defined at  $\;z\boldsymbol{=}t$ since given that $0<\upsilon<1$
\begin{align}
f(t) & \boldsymbol{=}\lim_{\upsilon\boldsymbol{\rightarrow} 1}\dfrac{1\boldsymbol{-}\upsilon^2}{\left[\left(t\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}
\nonumber\\
& \boldsymbol{=}\lim_{\upsilon\boldsymbol{\rightarrow} 1}\dfrac{1\boldsymbol{+}\upsilon}{\left(1\boldsymbol{-}\upsilon\right)^{\,1/2}\left[\left(1\boldsymbol{-}\upsilon\right)t^2\boldsymbol{+}\left(1\boldsymbol{+}\upsilon\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}\boldsymbol{=}\boldsymbol{+}\infty
\tag{05}\label{05}
\end{align}
We'll see now if our function fulfills a condition similar to \eqref{01b}. For $\boldsymbol{\varepsilon} \boldsymbol{>}0$ we have
\begin{align}
\int\limits_{\boldsymbol{t-\varepsilon}}^{\boldsymbol{t+\varepsilon}}\!\!\!f(z)\mathrm dz &\boldsymbol{=}\int\limits_{\boldsymbol{t-\varepsilon}}^{\boldsymbol{t+\varepsilon}}\!\!\!\lim_{\upsilon\boldsymbol{\rightarrow} 1}\dfrac{1\boldsymbol{-}\upsilon^2}{\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}\mathrm dz
\nonumber\\
&\boldsymbol{=}\lim_{\upsilon\boldsymbol{\rightarrow} 1}\int\limits_{\boldsymbol{t-\varepsilon}}^{\boldsymbol{t+\varepsilon}}\!\!\!\dfrac{1\boldsymbol{-}\upsilon^2}{\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}\mathrm dz
\nonumber\\
&\boldsymbol{=}\lim_{\upsilon\boldsymbol{\rightarrow} 1}\left[\dfrac{z\boldsymbol{-}\upsilon\,t}{r^2_{\boldsymbol{\perp}}\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,1/2}}\vphantom{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\right]_{z\boldsymbol{=}\boldsymbol{t-\varepsilon}}^{z\boldsymbol{=}\boldsymbol{t+\varepsilon}}
\nonumber\\
&\boldsymbol{=}\dfrac{1}{r^2_{\boldsymbol{\perp}}}\left[\lim_{\upsilon\boldsymbol{\rightarrow} 1}\dfrac{z\boldsymbol{-}\upsilon\,t}{\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,1/2}}\vphantom{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\right]_{z\boldsymbol{=}\boldsymbol{t-\varepsilon}}^{z\boldsymbol{=}\boldsymbol{t+\varepsilon}}
\nonumber\\
&\boldsymbol{=}\dfrac{1}{r^2_{\boldsymbol{\perp}}}\left[\dfrac{z\boldsymbol{-}t}{\boldsymbol{\vert}z\boldsymbol{-}t\boldsymbol{\vert}}\right]_{z\boldsymbol{=}\boldsymbol{t-\varepsilon}}^{z\boldsymbol{=}\boldsymbol{t+\varepsilon}} \boldsymbol{=}\dfrac{1}{r^2_{\boldsymbol{\perp}}}\left(\dfrac{\varepsilon}{\boldsymbol{\vert}\varepsilon\boldsymbol{\vert}}\boldsymbol{-}\dfrac{\boldsymbol{-}\varepsilon}{\boldsymbol{\vert}\boldsymbol{-}\varepsilon\boldsymbol{\vert}}\right)
\nonumber\\
&\boldsymbol{=}\dfrac{1}{r^2_{\boldsymbol{\perp}}}\dfrac{2\varepsilon}{\boldsymbol{\vert}\varepsilon\boldsymbol{\vert}}\boldsymbol{=}\dfrac{2}{r^2_{\boldsymbol{\perp}}}
\tag{06}\label{06}
\end{align}
In equation \eqref{06} the second equality is valid under the assumption that the limit and integral operators commute, while the third equality is due to the following indefinite integral

\begin{equation}
\int\dfrac{1}{\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,3/2}}\mathrm dz\boldsymbol{=}\dfrac{z\boldsymbol{-}\upsilon\,t}{\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\left[\left(z\boldsymbol{-}\upsilon\,t\right)^2\boldsymbol{+}\left(1\boldsymbol{-}\upsilon^2\right)r^2_{\boldsymbol{\perp}}\vphantom{\tfrac{a}{b}}\right]^{\,1/2}}\boldsymbol{+}\texttt{const.}
\tag{07}\label{07}
\end{equation}

By equations \eqref{04} and \eqref{06} we have proved equation \eqref{4 of paper}.

