Delta function indicated with an arrow, as well as different ways of defining the delta function as a limit of a distribution with fixed integral of 1.
For more Fourier analysis figures, please see the “fourier analysis” tag. These figures are used in Ben Kilminster’s lecture notes for PHY111.

delta_function-001.pngdelta_function-002.pngdelta_function-003.pngdelta_function-004.pngdelta_function-005.png

Edit and compile if you like:

% Author: Izaak Neutelings (February 2021)
\documentclass[border=3pt,tikz]{standalone}
\usepackage{tikz}
\usepackage{physics}
\usepackage{xcolor}

\tikzset{>=latex} % for LaTeX arrow head
\colorlet{myred}{red!85!black}
\colorlet{myblue}{blue!80!black}
\colorlet{mydarkred}{myred!80!black}
\colorlet{mydarkblue}{myblue!80!black}
\tikzstyle{xline}=[myblue,very thick]
\def\tick#1#2{\draw[thick] (#1) ++ (#2:0.1) --++ (#2-180:0.2)}
\def\N{80}

\begin{document}


% DELTA FUNCTION
\def\xmin{-0.7*\T} % min x axis
\def\xmax{1.7}     % max x axis
\def\ymin{-0.4}    % min y axis
\def\ymax{1.6}     % max y axis
\def\A{0.76*\ymax} % amplitude
\begin{tikzpicture}
  \message{^^JDelta function}
  \def\a{0.4*\xmax}
  \tick{0,\A}{0} node[left=-1,scale=0.9] {1};
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] (-\xmax,0) -- (\xmax+0.1,0) node[below=1,right=1] {$x$};
  \draw[xline,line cap=round] (-0.9*\xmax,0) -- (0.9*\xmax,0);
  \draw[xline,->] (0,0) --++ (0,\A) node[mydarkblue,below right=1] {$\delta(x)$};
  \draw[xline,thick,fill=white] (0,0) circle(0.05);
\end{tikzpicture}


% DELTA FUNCTION - shift
\begin{tikzpicture}
  \message{^^JDelta function - shift}
  \def\a{0.70*\xmax}
  \tick{-\a,\A}{0} node[left=-1,scale=0.9] {1};
  \tick{0,0}{90} node[below=-1,scale=0.9] {$a$};
  \draw[->,thick] (-\a,\ymin) -- (-\a,\ymax) node[left] {$y$};
  \draw[->,thick] (-\xmax,0) -- (\xmax+0.1,0) node[below=1,right=1] {$x$};
  \draw[xline,line cap=round] (-0.9*\xmax,0) -- (0.9*\xmax,0);
  \draw[xline,->] (0,0) -- (0,\A) node[mydarkblue,below right=1] {$\delta(x-a)$};
  \draw[xline,thick,fill=white] (0,0) circle(0.05);
\end{tikzpicture}


% STEP FUNCTION
\begin{tikzpicture}
  \message{^^JStep function}
  \def\A{0.76*\ymax} % amplitude
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] (-\xmax,0) -- (\xmax+0.1,0) node[below=1,right=1] {$x$};
  \tick{0,\A}{0} node[left=-1,scale=0.9] {1};
  \draw[xline,very thick,line cap=round]
    (-0.9*\xmax,0) -- (0,0)
    (0,\A) -- (0.9*\xmax,\A);
  \draw[mydarkblue,dashed,thin,line cap=round]
    (0,0) -- (0,\A);
  \fill[xline] (0,\A/2) circle(0.05);
  \draw[xline,thick,fill=white] (0,0) circle(0.05);
  \draw[xline,thick,fill=white] (0,\A) circle(0.05);
\end{tikzpicture}


% DELTA FUNCTION - RECTANGULAR FUNCTION
\begin{tikzpicture}
  \message{^^JDelta function - rectangular limit}
  \def\A{0.76*\ymax} % amplitude
  \def\T{0.15*\xmax} % width
  \fill[myblue!10] (-\T,0) rectangle (\T,\A);
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] (-\xmax,0) -- (\xmax+0.1,0) node[below=1,right=1] {$x$};
  \draw[xline,very thick,line cap=round]
    ( \T,\A) -- (-\T,\A) node[black,below=2,left=0,scale=0.9] {$\dfrac{1}{\epsilon}$}
    (-\T,0) -- ({-0.9*\xmax},0)
    ( \T,0) -- ({ 0.9*\xmax},0);
  \draw[mydarkblue,dashed,thin,line cap=round]
    (-\T,0) --++ (0,{\A})
    ( \T,0) --++ (0,{\A});
  \tick{-\T,0}{90} node[left=8,below=-3,scale=0.9] {\strut$-\epsilon/2$};
  \tick{ \T,0}{90} node[right=5,below=-3,scale=0.9] {\strut$\epsilon/2$};
  \fill[xline] (-\T,\A) circle(0.05);
  \fill[xline] ( \T,\A) circle(0.05);
  \draw[xline,thick,fill=white] (-\T,0) circle(0.05);
  \draw[xline,thick,fill=white] ( \T,0) circle(0.05);
\end{tikzpicture}


% DELTA FUNCTION - GAUSSIAN FUNCTION
\begin{tikzpicture}
  \message{^^JDelta function - Gaussian limit}
  \def\A{0.76*\ymax} % amplitude
  \def\T{0.14*\xmax} % width
  \fill[myblue!10,samples=\N,variable=\x,domain=-0.9*\xmax:0.9*\xmax]
    plot(\x,{\A*exp(-(\x/(\T))^2/2)});
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] (-\xmax,0) -- (\xmax+0.1,0) node[below=1,right=1] {$x$};
  \draw[xline,very thick,smooth,samples=\N,variable=\x,domain=-0.9*\xmax:0.9*\xmax]
    plot(\x,{\A*exp(-(\x/(\T))^2/2)});
  \node[mydarkblue,right=0] at (0.1*\xmax,0.67*\ymax)
    {$\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}}$};
  \draw[mydarkblue,dashed,thin,line cap=round]
    (-\T,0) |- (\T,{\A*exp(-1/2)}) -- (\T,0);
  \tick{-\T,0}{90} node[left=5,below=-4,scale=0.9] {\strut$-\sigma$};
  \tick{ \T,0}{90} node[right=2,below=-4,scale=0.9] {\strut$\sigma$};
\end{tikzpicture}


\end{document}

Click to download: delta_function.texdelta_function.pdf
Open in Overleaf: delta_function.tex

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