# Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function

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## Abstract

**:**

## 1. Introduction

## 2. Derivation of the Definite Integral of the Contour Integral

## 3. Derivation of the Contour Integral in Terms of the Lerch Function

#### 3.1. The Lerch Function

#### 3.2. Derivation of the Infinite Sum of the Contour Integral

#### 3.2.1. Derivation of the First Contour Integral

#### 3.2.2. Derivation of the Second Contour Integral

## 4. Definite Integral in Terms of the Lerch Function

## 5. Evaluation of Special Cases of Definite INTEGRALS

#### 5.1. Special Case 1

#### 5.2. Special Case 2

## 6. Derivation of Entry 3.514.4 in Gradshteyn, I.S.; et al.

## 7. Derivation of Entry (2.3.1.19) in Yu, A.; et al.

## 8. Derivation of a New Entry for Table 3.514 in Gradshteyn, I.S.; et al.

## 9. Definite Integral in Terms of the Hurwitz Zeta Function

## 10. Definite Integral in Terms of the Log-Gamma $log(\Gamma (x\left)\right)$ and Harmonic Number ${H}_{k}$ Functions

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 11. Derivation of Hyperbolic and Algebraic Forms

**Example**

**5.**

**Example**

**6.**

## 12. Discussion

## 13. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Reynolds, R.; Stauffer, A.
Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function. *Sci* **2021**, *3*, 37.
https://doi.org/10.3390/sci3040037

**AMA Style**

Reynolds R, Stauffer A.
Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function. *Sci*. 2021; 3(4):37.
https://doi.org/10.3390/sci3040037

**Chicago/Turabian Style**

Reynolds, Robert, and Allan Stauffer.
2021. "Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function" *Sci* 3, no. 4: 37.
https://doi.org/10.3390/sci3040037