Heaviside function in the form of an integral I am currently reading Optimal storage properties of neural network models by E. Gardner. (DOI 10.1088/0305-4470/21/1/031)
In appendix 1, the Heaviside function is expressed in integral form eq A1.1
$$\theta\left(\frac{1}{\sqrt{N}}\sum_jS^{\mu}_iS^{\mu}_jJ^{\alpha}_{ij}-K\right)=\frac{1}{2\pi}\int^{\infty}_{K}d\lambda^{\alpha}_{\mu}\int^{\infty}_{-\infty}dx^{\alpha}_{\mu}\exp\left(ix^{\alpha}_{\mu}(\lambda^{\alpha}_{\mu}-\frac{1}{\sqrt{N}}\sum_jS^{\mu}_iS^{\mu}_jJ^{\alpha}_{ij})\right).$$
I do not understand how this form is achieved, nor can I check its validity.
 A: $\newcommand{\bl}[1]{\boldsymbol{#1}} 
\newcommand{\e}{\bl=}
\newcommand{\p}{\bl+}
\newcommand{\m}{\bl-}
\newcommand{\mr}[1]{\mathrm {#1}}
\newcommand{\gr}{\bl>}
\newcommand{\les}{\bl<}
\newcommand{\greq}{\bl\ge}
\newcommand{\leseq}{\bl\le}
\newcommand{\plr}[1]{\left(#1\right)}
\newcommand{\blr}[1]{\left[#1\right]}
\newcommand{\tl}[1]{\tag{#1}\label{#1}}$
Changing the notation
\begin{equation}
x^\alpha_\mu\bl\rightarrow x\,,\qquad \lambda^\alpha_\mu\bl\rightarrow \lambda,\qquad \frac{1}{\sqrt{N\vphantom{N^2}}}\sum_jS^{\mu}_iS^{\mu}_jJ^{\alpha}_{ij}\bl\rightarrow s
\tl{01}
\end{equation}
the equation is expressed in simple form
\begin{equation}
\theta\plr{s\m K}\e\!\!\int\limits_K^{\p \bl\infty}\!\!\mr d\lambda\plr{\dfrac{1}{2\pi}\int\limits_{\m \bl\infty}^{\p \bl\infty}\!\!\mr dx\, e^{\m i\plr{s\m\lambda}x}}
\tl{02}
\end{equation}
From the integral definition of the Dirac $\:\delta\m$function
\begin{equation}
\delta\plr{k}\e\dfrac{1}{2\pi}\int\limits_{\m \bl\infty}^{\p \bl\infty}\!\!e^{\boldsymbol{-}ikx}\mathrm dx
\tl{03}
\end{equation}
we have
\begin{equation}
\dfrac{1}{2\pi}\int\limits_{\m \bl\infty}^{\p \bl\infty}\!\!\mr dx\,e^{\m i\plr{s\m\lambda}x}\e \delta\plr{s\m\lambda}\e\delta\plr{\lambda\m s}
\tl{04}
\end{equation}
So
\begin{align}
&\int\limits_K^{\p \bl\infty}\!\!\mr d\lambda\plr{\dfrac{1}{2\pi}\int\limits_{\m \bl\infty}^{\p \bl\infty}\!\!\mr dx\, e^{\m i\plr{s\m\lambda}x}}\e \int\limits_K^{\p \bl\infty}\!\!\mr d\lambda\,\delta\plr{\lambda\m s}\e
\nonumber\\
& \int\limits_{K\m s}^{\p \bl\infty}\!\!\mr d\plr{\lambda\m s}\,\delta\plr{\lambda\m s}\e \int\limits_{y_0\e K\m s}^{\p \bl\infty}\!\!\mr dy\,\delta\plr{y}\e\left.
\begin{cases}
0 \quad\texttt{if} \quad y_0\e K\m s\gr 0\\
1 \quad\texttt{if} \quad y_0\e K\m s\leq 0
\end{cases}
\right\}\e
\nonumber\\
&\left.
\begin{cases}
0 \quad\texttt{if} \quad s\m K\les 0\\
1 \quad\texttt{if} \quad s\m K\greq 0
\end{cases}
\right\}\e \theta\plr{s\m K}
\tl{05}
\end{align}
