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Mathematica Codes

Plotting the image of the unit disk under exponential function


f[z_] := Exp[z]

ff[r_, t_] := f[r  Exp[I t]];

u[r_, t_] := Re[ff[r, t]];

v[r_, t_] := Im[ff[r, t]];

ParametricPlot[{u[r, t], v[r, t]}, {r, 0, 1}, {t, 0, 2 Pi}, PlotStyle -> Red, BoundaryStyle -> Blue, MeshStyle -> Black, PlotPoints -> 100, Mesh -> 10, AxesOrigin -> {0, 0}]

Plotting the images of the unit circle under 4 functions: Sqrt[1 + z], Exp[z], Sin[z] and 2/(1+Exp[-z])

ParametricPlot[{{Re[Sqrt[1 + Exp[I t]]], Im[Sqrt[1 + Exp[I t]]]}, {Re[Exp[Exp[I t]]], Im[Exp[Exp[I t]]]}, {Re[1 + Sin[Exp[I t]]], Im[1 + Sin[Exp[I t]]]}, {Re[2/(1 + Exp[-Exp[I t]])], Im[2/(1 + Exp[-Exp[I t]])]},}, {t, -Pi, Pi}, PlotStyle -> {Red, Blue, Green, Yellow}]

Illustration of the sharpness of the radius constant in Corollary 3.1 of the research article [Mendiratta, Rajni; Nagpal, Sumit; Ravichandran, V. On a subclass of strongly starlike functions associated with exponential function. Bull. Malays. Math. Sci. Soc. 38 (2015), no. 1, 365--386. MR3394060]


l[z_] := z/(1 - z)

f[z_] := z l'[z]/l[z];

ff[r_, t_] := f[r  Exp[I t]];

u[r_, t_] := Re[ff[r, t]];

v[r_, t_] := Im[ff[r, t]];

Show[ParametricPlot[{u[r, t], v[r, t]}, {r,0, (Exp[1] - 1)/(Exp[1])}, {t, 0, 2 Pi}, PlotStyle -> Gray, BoundaryStyle -> Black, MeshStyle -> Black, PlotPoints -> 100, Mesh -> 10, AxesOrigin -> {0, 0}], ParametricPlot[{Re[Exp[Exp[I t]]], Im[Exp[Exp[I t]]]}, {t, 0, 2 Pi}],PlotRange -> Full]

LaTeX Codes

Making Mark-Strohhacker Implications in LaTeX

\documentclass[12pt, a4paper]{amsart}

\usepackage{tikz}

\usetikzlibrary{arrows.meta}

\renewcommand{\Re}{\textrm{Re}}

\begin{document}

\begin{figure}[h]

\begin{tikzpicture}

\node[right] at (0,4) {$\Re \dfrac{zf''(z)}{f'(z)}+1>0$};

\node[right] at (5,6) {$\Re \dfrac{zf'(z)}{f(z)}>\dfrac{1}{2}$};

\node[right] at (9.5,4) {$\Re \dfrac{f(z)}{z}>\dfrac{1}{2}$};

\node[right] at (5,2) {$\Re \sqrt{f'(z)}>\dfrac{1}{2}$};

\draw[-Implies,double distance=3pt] (3.75,4.75) -- (5,5.5);

\draw[-Implies,double distance=3pt] (3.75,3.25) -- (5,2.5);

\draw[-Implies,double distance=3pt] (8,5.5) -- (9.25,4.75);

\draw[-Implies,double distance=3pt] (8,2.5) -- (9.25,3.25);

\end{tikzpicture}

\caption{Mark-Strohhacker Implication}

\end{figure}

\end{document}

Making graphs in LaTeX

\documentclass[12pt, a4paper]{amsart}

\usepackage{tikz}

\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}

\begin{axis}[xmin=0, xmax=2.5*pi,

ymin=-2, ymax=2,

axis lines=middle,

xtick={0, pi/2, pi, 3*pi/2, 2*pi},

xticklabels={$0$, $\pi/2$, $\pi$, $3\pi/2$, $2\pi$}, samples=1000, domain=0: 2*pi,

xmajorgrids=true,

grid style=dashed,

ticklabel style={font=\tiny}]

\addplot[red]{sin(deg(x))}

node[right, pos=0.3]{$\sin x$};

\addlegendentry{$\sin x$}

\addplot[blue]{cos(deg(x))}

node[right, pos=0.9]{$\cos x$}; 

\addlegendentry{$\cos x$}

\end{axis}

\end{tikzpicture}

\end{document}

Making disks in LaTeX

\documentclass[12pt, a4paper]{amsart}

\usepackage{tikz}

\renewcommand{\Re}{\textrm{Re}}

\renewcommand{\Im}{\textrm{Im}}

\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}

\pgfplotsset{width=10cm}

\begin{axis}[font=\tiny,

xmin=-5, xmax=5,

ymin=-5, ymax=5,

xtick={-4,-3,-2,-1,0,1,2,3,4},

ytick={-4,-3,-2,-1,0,1,2,3,4},

xticklabels={$-4$,$-3$,$-2$,$-1$, $0$,$1$,$2$,$3$,$4$},

yticklabels={$-4i$,$-3i$,$-2i$,$-1i$, $0i$,$1i$,$2i$,$3i$,$4i$},

axis equal,

axis lines=middle,

xlabel=$\Re(z)$,

ylabel=$\Im(z)$,

disabledatascaling]

\fill [opacity=0.3] (1,-1) circle [radius=3];

\node [below right] at (1,-1) {$|z-1+i| \leq 3$};

\end{axis}

\end{tikzpicture}

\end{document}

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