# On the Connection Problem for Painlevé Differential Equation in View of Geometric Function Theory

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

## 3. Connection Bounds

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

- Let$\rho (\zeta )=\frac{1+\zeta}{1-\zeta}=1+2{\zeta}^{2}+2{\zeta}^{3}+\dots $then$|{\chi}_{2}|\le \frac{2}{3},\phantom{\rule{1.em}{0ex}}|{\chi}_{3}|\le \frac{{\rho}_{2}+{\rho}_{1}^{2}/3}{8}=0.416.$
- Let$\rho (\zeta )={\left(\right)}^{\frac{1+\zeta}{1-\zeta}}0.5$then$$|{\chi}_{2}|\le \frac{1}{3},\phantom{\rule{1.em}{0ex}}|{\chi}_{3}|\le \frac{0.5+1/3}{8}=0.104.$$

**Corollary**

**1.**

## 4. Geometric Behaviors

**Definition**

**3.**

**Lemma**

**1.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**2.**

**Theorem**

**3.**

**Proof.**

**Example**

**2.**

- Let$q=1$, we have$$\int \phantom{\rule{0.166667em}{0ex}}|{e}^{i\theta}+\sqrt{{({e}^{i\theta})}^{2}+1}|d\theta =-i({e}^{i\theta}+\sqrt{1+{e}^{2i\theta}}-tan{h}^{-1}(\sqrt{1+{e}^{2i\theta})})+constant$$
- Let$q=2$then we get$$\int \phantom{\rule{0.166667em}{0ex}}|{e}^{i\theta}+\sqrt{{({e}^{i\theta})}^{2}+1}{|}^{2}d\theta =\theta -i{e}^{2i\theta}-i{e}^{i\theta}\sqrt{1+{e}^{2i\theta}}-i{sinh}^{-1}({e}^{i\theta})+constant.$$

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**4.**

**Proof.**

## 5. Symmetric Solution

**Definition**

**6.**

**Lemma**

**3.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Example**

**3.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The asymptotic expansions of $sin\left(\right)open="("\; close=")">\frac{\zeta}{1-\zeta}$ and $sin\left(\right)open="("\; close=")">\frac{\zeta}{{(1-\zeta )}^{2}}$, respectively.

**Figure 2.**The complex plane, Riemann surface and the asymptotic expansions of $(\zeta +\sqrt{{\zeta}^{2}+1})$, respectively.

**Figure 4.**The behavior of $\mathsf{\Phi}(\zeta )=\frac{1+{\zeta}^{2}}{1-{\zeta}^{2}}$ with a symmetric domain for $|\zeta |,1$.

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**MDPI and ACS Style**

Ibrahim, R.W.; Elobaid, R.M.; Obaiys, S.J.
On the Connection Problem for Painlevé Differential Equation in View of Geometric Function Theory. *Mathematics* **2020**, *8*, 1198.
https://doi.org/10.3390/math8071198

**AMA Style**

Ibrahim RW, Elobaid RM, Obaiys SJ.
On the Connection Problem for Painlevé Differential Equation in View of Geometric Function Theory. *Mathematics*. 2020; 8(7):1198.
https://doi.org/10.3390/math8071198

**Chicago/Turabian Style**

Ibrahim, Rabha W., Rafida M. Elobaid, and Suzan J. Obaiys.
2020. "On the Connection Problem for Painlevé Differential Equation in View of Geometric Function Theory" *Mathematics* 8, no. 7: 1198.
https://doi.org/10.3390/math8071198