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!12,draw=black!40,line width=.35pt] (0.00000,3.90000) rectangle ++(3.40000,0.70000);
3 \node[] at (1.70000,4.25000) {function};
4 \draw[fill=black!12,draw=black!40,line width=.35pt] (3.40000,3.90000) rectangle ++(1.60000,0.70000);
5 \node[] at (4.20000,4.25000) {$1$};
6 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.00000,3.90000) rectangle ++(2.10000,0.70000);
7 \node[] at (6.05000,4.25000) {$z^2/2!$};
8 \draw[fill=black!12,draw=black!40,line width=.35pt] (7.10000,3.90000) rectangle ++(2.10000,0.70000);
9 \node[] at (8.15000,4.25000) {$z^4/4!$};
10 \draw[fill=black!12,draw=black!40,line width=.35pt] (9.20000,3.90000) rectangle ++(2.10000,0.70000);
11 \node[] at (10.25000,4.25000) {$z^6/6!$};
12 \draw[fill=black!0,draw=black!40,line width=.35pt] (0.00000,3.05000) rectangle ++(3.40000,0.80000);
13 \node[] at (1.70000,3.45000) {$\sec z$};
14 \draw[fill=black!0,draw=black!40,line width=.35pt] (3.40000,3.05000) rectangle ++(1.60000,0.80000);
15 \node[] at (4.20000,3.45000) {$1$};
16 \draw[fill=black!8,draw=black!40,line width=.35pt] (5.00000,3.05000) rectangle ++(2.10000,0.80000);
17 \node[] at (6.05000,3.45000) {$1$};
18 \draw[fill=black!0,draw=black!40,line width=.35pt] (7.10000,3.05000) rectangle ++(2.10000,0.80000);
19 \node[] at (8.15000,3.45000) {$5$};
20 \draw[fill=black!8,draw=black!40,line width=.35pt] (9.20000,3.05000) rectangle ++(2.10000,0.80000);
21 \node[] at (10.25000,3.45000) {$61$};
22 \draw[fill=black!0,draw=black!40,line width=.35pt] (0.00000,2.15000) rectangle ++(3.40000,0.80000);
23 \node[] at (1.70000,2.55000) {$\operatorname{sech}z$};
24 \draw[fill=black!0,draw=black!40,line width=.35pt] (3.40000,2.15000) rectangle ++(1.60000,0.80000);
25 \node[] at (4.20000,2.55000) {$1$};
26 \draw[fill=black!8,draw=black!40,line width=.35pt] (5.00000,2.15000) rectangle ++(2.10000,0.80000);
27 \node[] at (6.05000,2.55000) {$-1$};
28 \draw[fill=black!0,draw=black!40,line width=.35pt] (7.10000,2.15000) rectangle ++(2.10000,0.80000);
29 \node[] at (8.15000,2.55000) {$5$};
30 \draw[fill=black!8,draw=black!40,line width=.35pt] (9.20000,2.15000) rectangle ++(2.10000,0.80000);
31 \node[] at (10.25000,2.55000) {$-61$};
32 \draw[fill=black!0,draw=black!40,line width=.35pt] (0.00000,1.25000) rectangle ++(3.40000,0.80000);
33 \node[] at (1.70000,1.65000) {$z/\sinh z$};
34 \draw[fill=black!0,draw=black!40,line width=.35pt] (3.40000,1.25000) rectangle ++(1.60000,0.80000);
35 \node[] at (4.20000,1.65000) {$1$};
36 \draw[fill=black!8,draw=black!40,line width=.35pt] (5.00000,1.25000) rectangle ++(2.10000,0.80000);
37 \node[] at (6.05000,1.65000) {$- \frac{1}{3}$};
38 \draw[fill=black!0,draw=black!40,line width=.35pt] (7.10000,1.25000) rectangle ++(2.10000,0.80000);
39 \node[] at (8.15000,1.65000) {$\frac{7}{15}$};
40 \draw[fill=black!8,draw=black!40,line width=.35pt] (9.20000,1.25000) rectangle ++(2.10000,0.80000);
41 \node[] at (10.25000,1.65000) {$- \frac{31}{21}$};
42 \node[] at (5.65000,-0.15000) {$E_{2,1}(z)=\sinh z=z\Phi_{2,1}(z^2),\qquad C_{2,1}(0)=1$};
43 \end{tikzpicture}

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