Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round,line join=round]
2 \draw[fill=black!4,draw=black!45] (2.8,2.3) circle (1.6);
3 \draw[->] (0.65000,2.30000) -- (5.20000,2.30000);
4 \draw[->] (2.80000,0.30000) -- (2.80000,4.40000);
5 \draw[fill=black] (2.80000,2.30000) circle (0.055);
6 \node[] at (2.80000,4.65000) {fixed-log domain about $0$};
7 \draw[fill=white,line width=1pt] (4.40000,2.30000) circle (0.09);
8 \node[] at (5.10000,1.45000) {$\rho=\pi/2$};
9 \draw[->,line width=1.1pt] (4.88,2.3) arc[start angle=0,end angle=330,radius=.48];
10 \node[] at (2.80000,0.10000) {the loop is not inside the disk};
11 \node[] at (9.20000,4.10000) {$D(z)\sim D^{\prime}(\rho)(z-\rho)$};
12 \node[] at (9.20000,2.90000) {$\log D\longmapsto\log D+2\pi i$};
13 \node[] at (9.20000,1.65000) {$D^{-r}\longmapsto e^{-2\pi i r}D^{-r}$};
14 \node[] at (9.20000,0.40000) {$r=1/2:\ -1\qquad r\in\mathbb Z:\ 1$};
15 \end{tikzpicture}

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