# Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Centered Polygonal Lacunary Functions

**Lemma**

**1.**

**Lemma**

**2.**

**Definition**

**1.**

**Theorem**

**1.**

## 3. The $\mathit{p}$–Sequences

**Definition**

**2.**

**Theorem**

**2.**

**Proof.**

## 4. Symmetry Angle Spaces

## 5. Cyclic Decomposition

## 6. Parametric Curves

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Conjecture**

**1.**

**Remark**

**1.**

## 7. Whole Sphere Mapping

**Theorem**

**5.**

**Proof.**

**Conjecture**

**2.**

**Remark**

**2.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Conjecture**

**3.**

**Remark**

**3.**

**Theorem**

**6.**

**Proof.**

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A particularly illustrative way to present graphs of $\mathfrak{L}({C}^{\left(k\right)};z)$. The representation shown here is especially useful for this work. The contour plot is truncated at the unity level set (blue shading represents low values and red shading represents high values). The top left panel shows the example of $\mathfrak{L}({C}^{\left(3\right)};z)$ where a plot of $\left|{f}_{16}\left(z\right)\right|$. The top right panel shows a superposition of the contour plot and a three-dimensional rendering. The truncated contour plot more clearly exposes the true rotational symmetry of the centered polygonal lacunary functions. The bottom left graph shows the case of $\mathfrak{L}({T}_{n};z)$, where $T\left(n\right)$ are the well-known triangular numbers, again for $\left|{f}_{16}\left(z\right)\right|$. Despite the intimate relationship between the centered polygonal numbers and the triangular numbers, the plots are strikingly different. The bottom right panel shows an unshaded contour plot of the same function shown in the left panel of Figure 1. The superimposed black lines indicate the symmetry angles. The first 15 symmetry angles are shown (see text for details).

**Figure 2.**Cyclic decompositions for the centered polygonal lacunary functions along three of the line segments shown in the bottom right panel of Figure 1, that is, $k=3$. The first 40 ${f}_{j}$ are shown. The top row shows $|{f}_{40p}(\rho {e}^{\frac{i\pi}{3p}})|$: left panel $p=1$, middle panel $p=2$, right panel $p=3$. The bottom row focuses on the $p=1$ case in more detail. The left and middle panels show a sequential blow up near the natural boundary of the top left graph (note the displayed domain on the $\rho $ axis). For better clarity, the first 10 ${f}_{j}$ are not shown in the left panel and the first 20 ${f}_{j}$ are not shown in the middle panel. Finally, the bottom right panel shows the real (blue) and imaginary (red) parts of ${f}_{40}(\rho {e}^{\frac{i\pi}{3}})$, that is, $k=3$, $p=1$.

**Figure 4.**Parametric curves, ${\mathsf{P}}^{\left(3\right)}(\rho ;p)$ from Equation (12), of ${f}_{N}\left(z\right)$ for four different values of k read left-to-right, top-to-bottom: $k=1$, $k=2$, $k=4$, $k=8$. Shown are the first 10 values of p. The case of $p=1$ has no interior points and is directed at an angle equal to ${\alpha}_{1}$ in ${\mathbb{R}}^{2}$. Increasing values of p lead to closed curves which are bigger and have greater interior area. As $\rho $ goes from 0 to 1 the curve is traversed in a counterclockwise direction.

**Figure 5.**Top: Parametric plot ${\mathsf{P}}^{\left(3\right)}(\rho ;p)$ (black curve) superimposed with red vectors indicating the “velocity” along the curve. One notices a modest “acceleration” until the curve turns back towards the origin whereupon the acceleration is markedly increased. Bottom left: the arclength (ordinate) versus $\rho $ (abscissa) for ${\mathsf{P}}^{\left(3\right)}(\rho ;p)$. Bottom right: Arclength of ${\mathsf{P}}^{\left(k\right)}(\rho ;p)$ for $p=1,2,\dots ,30$ (black dots) associated with the parametric plots shown in Figure 4 fitted to $A\sqrt{p}+c$ (orange curve). The top curve is for $k=1$ and the bottom curve is for $k=8$. Fit parameters $(A,c)$ for $k=1$, $k=2$, $k=4$, $k=8$ respectively: $(1.7880,-1.1038)$, $(1.7571,-0.7926)$, $(1.7175,-0.5052)$, $(1.6794,-0.2701)$.

**Figure 6.**Plot of area for $1\le p\le 17$ and $1\le k\le 5$. Where the lowest set is the area of $k=1$ and each successively higher line corresponds to the next greatest k value. The area for each p value approaches to a distinct line for each k value.

**Figure 7.**The whole sphere mapping of $\overline{\mathcal{D}}$ onto ${S}^{2}$ (see text for the equations of the map). The mapping is that of the centered polygonal lacunary function shown in Figure 1 under $\widehat{S}$. Two different viewpoints of the same function ( $|{f}_{16}^{\left(3\right)}(\varphi ,\theta )|$) are shown. The left panel shows a “front” view such that the north pole $(0,0,1)$ is located directly on top and the south pole $(0,0,-1)$ directly on the bottom. The right panel shows the “bottom” view such that the south pole is directly in the center of the image. The unit circle maps to the single point at the south pole.

**Figure 9.**${I}_{1}$ versus k (dots: $\left|{I}_{1}\right|$ - black, $\mathrm{Re}\left[{I}_{1}\right]$ - blue, $\mathrm{Im}\left[{I}_{1}\right]$ - red). The curves arise from Equation (30) in Corollary 2.

**Figure 10.**Left Panel: ${L}_{p1}$ versus p (dots: $\left|{L}_{p1}\right|$ - black, $\mathrm{Re}\left[{L}_{p1}\right]$ - blue, $\mathrm{Im}\left[{L}_{p1}\right]$ - red) for the case of $k=1$. The curves arise from Equation (30) in Corollary 2. Right Panel: $\frac{k}{\pi}{L}_{\infty 1}$ versus k ($\left|\frac{k}{\pi}{L}_{\infty 1}\right|$ - black, $\mathrm{Re}\left[\frac{k}{\pi}{L}_{\infty 1}\right]$ - blue, $\mathrm{Im}\left[\frac{k}{\pi}{L}_{\infty 1}\right]$ - red). The dashed lines represent the ${\mathrm{lim}}_{k\to \infty}\frac{k}{\pi}{L}_{\infty 1}=1-i$.

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**MDPI and ACS Style**

Mork, L.K.; Sullivan, K.; Ulness, D.J.
Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space. *Mathematics* **2020**, *8*, 568.
https://doi.org/10.3390/math8040568

**AMA Style**

Mork LK, Sullivan K, Ulness DJ.
Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space. *Mathematics*. 2020; 8(4):568.
https://doi.org/10.3390/math8040568

**Chicago/Turabian Style**

Mork, Leah K., Keith Sullivan, and Darin J. Ulness.
2020. "Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space" *Mathematics* 8, no. 4: 568.
https://doi.org/10.3390/math8040568