# Integration of a Generalized Ratio of Polynomials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. Denominator of Linear Total Exponentiation

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

## 3. Denominator of Squared Total Exponentiation

## 4. Denominator of Arbitrary Total Exponentiation $\mathit{r}$

## 5. Spanning Multiple Poles in the Integration Range

## 6. Closing Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Numerical validation for a linear exponentiated denominator $(r=1)$. The roots are a mix of purely real and complex ${x}_{1,2,3}=\{1/2,1-i,1+i\}$. The singular point at $x=0.5$ was manually removed to avoid large divergences in the visualization.

**Figure 2.**Agreement between the closed-form solution shown in Equation (23), and the numerical integration of the polynomial ratio given in the plot title.

**Figure 3.**Plot of the closed-form solution shown in Figure 2 for increasing error at the location of the zeros.

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Brandsema, M.J.; Brocker, D.E.
Integration of a Generalized Ratio of Polynomials. *Mathematics* **2021**, *9*, 2579.
https://doi.org/10.3390/math9202579

**AMA Style**

Brandsema MJ, Brocker DE.
Integration of a Generalized Ratio of Polynomials. *Mathematics*. 2021; 9(20):2579.
https://doi.org/10.3390/math9202579

**Chicago/Turabian Style**

Brandsema, Matthew J., and Donovan E. Brocker.
2021. "Integration of a Generalized Ratio of Polynomials" *Mathematics* 9, no. 20: 2579.
https://doi.org/10.3390/math9202579