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\documentclass[12pt,a4paper]{amsbook}
%\documentclass[border=2mm]{standalone}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{appendix}
\usepackage{cancel}
\usepackage{url}
\usepackage{bm} %bold math

\usepackage{tikz,tkz-euclide}
\usetikzlibrary{positioning}
\usetikzlibrary{patterns}
\usepackage{siunitx}
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\usepackage{subcaption}
%\usepackage{pgfplots}
%\usepackage{pgfmath}
%\usetikzlibrary{calc}

%For GeoGebra pictures

\usepackage{pgf,tikz,pgfplots}
\pgfplotsset{compat=1.15}
\usepackage{mathrsfs}
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\pagestyle{empty}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\R}{\mathbb{R}} % mathbb-fonts
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Chat}{{\overline{\mathbb{C}}}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\D}{\mathbb{D}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\HH}{\mathbb{H}}%upper half plane
\def\stackrel#1#2{\mathrel{\mathop{#2}\limits^{#1}}} %stacking expressions

\DeclareMathOperator{\Real}{Re}
\DeclareMathOperator{\Imag}{Im}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}
\thispagestyle{empty}
\hspace{-0.5cm}
%hyperbolic densities of various domains
\begin{figure}
\hspace{-0.5cm}
\begin{tikzpicture}[>=stealth,scale=0.9]
\draw (0,2.5) node[scale=1.2]{$\rho_{\mathbb{D}}(z)=\frac{1}{1-|z|^2}$};
\draw (8,2.5) node[scale=1.2]{$\rho_{\mathbb{H}}(z)=\frac{1}{2\Imag(z)}$};
\draw (11,-4) node[scale=1.2]{$\rho_L(z)=\frac{1}{2\cos \Imag(z)}$};
\draw (-1.5,-3.5) node[scale=1.2]{$\rho_S(z)=\frac{\pi}{2\lambda\sin(\frac{\pi}{\lambda} \Imag(z))}$};
\draw (-1,-15) node[scale=1.2]{$\rho_{A(0,1,e^{\lambda})}(z)=\frac{\pi}{2\lambda|z|\sin(\frac{\pi}{\lambda}\log|z|)}$};
\draw (11,-8.5) node[scale=1.2]{$\rho_{A(0,1,\infty)}(z)=\frac{1}{2|z|\log|z|}$};
\draw (11,-17.5) node[scale=1.2]{$\rho_{A(0,0,1)}(z)=\frac{1}{2|z|\log\frac{1}{|z|}}$};

\draw[->] (2,0.5) -- node[above]{$a(z)=i\frac{1+z}{1-z}$} (4,0.5);
\draw[->] (8,-1.5) -- node[right]{$b(z)=\log(-iz)$} (8,-4);
\draw[->] (5,-4.5) -- node[above]{$c(z)=(z+i\frac{\pi}{2})\frac{\lambda}{\pi}$} (3,-4.5);
\draw[->] (-1,-6.5) -- node[right]{$d(z)=e^{iz}$} (-1,-8.5);
\draw[->] (3,-10) -- node[above]{$\lambda\to\infty$} (5,-10);
\draw[->] (9,-13.2) -- node[right]{$z\mapsto\frac{1}{z}$} (9,-14.8);
\draw[->] (1,-4) -- node[above,sloped]{$\lambda\to\infty$} (5,-1.5);

%unit disc
\begin{scope}
\filldraw[fill=gray] (0,0) circle (1cm);
\draw[->] (-1.5,0) -- (1.5,0);
\draw[->] (0,-1.5) -- (0,1.5);
\end{scope}

%upper half plane
\begin{scope}[xshift=8cm, yshift=-1cm]
%\filldraw[bottom color=gray, top color=white, draw=none]
%\filldraw[pattern=south west lines, draw=none]
%(-3,0) -- (3,0)
%-- (3,2) -- (-3,2) -- cycle;
\draw[pattern=north west lines, pattern color=gray, draw=none] (-3,0) rectangle (3,2);
\draw[->] (-4,0) -- (4,0);
%\draw[->] (0,-1.5) -- (0,1.5);
\end{scope}

%strip of height pi
\begin{scope}[yshift=-6cm,xshift=8cm,scale=0.5]
\filldraw[fill=gray]
(-5,-1.57) -- (5,-1.57)
-- (5,1.57) -- (-5,1.57) -- cycle;
\draw[->] (-7,0) -- (7,0);
\filldraw[fill=black] (0,-1.57) circle (5pt);
\filldraw[fill=black] (0,1.57) circle (5pt);
\draw (0,-1.57) node[below right]{$-i\frac{\pi}{2}$};
\draw (0,1.57) node[above right]{$i\frac{\pi}{2}$};
%\draw[->] (0,-1.5) -- (0,1.5);
\end{scope}

%strip of height lambda
\begin{scope}[yshift=-6cm,scale=0.5]
\filldraw[fill=gray]
(-5,0) -- (5,0)
-- (5,2) -- (-5,2) -- cycle;
\draw[->] (-7,0) -- (7,0);
%\filldraw[fill=black] (5,-1.57) circle (5pt);
\filldraw[fill=black] (0,2) circle (5pt);
%\draw (5,-1.57) node[right]{$-\frac{\pi}{2}$};
\draw (0,2) node[above right]{$i\lambda$};
%\draw[->] (0,-1.5) -- (0,1.5);
\end{scope}

%annulus
\begin{scope}[xshift=0cm,yshift=-11cm,scale=1]
\path [draw=none,fill=gray,even odd rule] (0,0) circle (1) (0,0) circle (0.5);
\filldraw[fill=black] (0.5,0) circle (2.5pt);
\filldraw[fill=black] (1,0) circle (2.5pt);
\draw (0.5,0) node[below]{$1$};
\draw (1,0) node[below right]{$e^{\lambda}$};
\draw[->] (-2.5,0) -- (2.5,0);
\draw[->] (0,-2.5) -- (0,2.5);
\end{scope}

%infinite-annulus
\begin{scope}[xshift=8cm,yshift=-11cm,scale=0.5]
\scope
\clip (-4,-4) rectangle (4,4);
\path [pattern=north west lines, pattern color=black,draw=none,even odd rule] (0,0) circle (6) (0,0) circle (1);
\endscope
\filldraw[fill=black] (1,0) circle (2.5pt);
%\filldraw[fill=black] (2,0) circle (2.5pt);
\draw (1,0) node[below right]{$1$};
%\draw (2,0) node[below right]{$e^{\lambda}$};
\draw[->] (-5,0) -- (5,0);
\draw[->] (0,-5) -- (0,5);
\end{scope}

%punctured disc
\begin{scope}[xshift=8cm,yshift=-16cm,scale=1]
\path [draw=none,fill=gray,even odd rule] (0,0) circle (1) (0,0) circle (0.1);
\filldraw[fill=black] (1,0) circle (2.5pt);
%\filldraw[fill=black] (2,0) circle (2.5pt);
\draw (1,0) node[below right]{$1$};
%\draw (2,0) node[below right]{$e^{\lambda}$};
\draw[->] (-1.5,0) -- (1.5,0);
\draw[->] (0,-1.5) -- (0,1.5);
\end{scope}
\end{tikzpicture}
\caption{Hyperbolic densities of various domains.}
\end{figure}

\end{document}