# Hyperbolic Functions

• João Pedro Morais
• Svetlin Georgiev
• Wolfgang Sprößig
Chapter

## Abstract

After bringing together various results mentioned before, in this chapter we introduce the quaternion hyperbolic functions, whose study will require us to master a new situation. Since the quaternion exponential function agrees with the real and complex exponential function of real and complex arguments, it follows that the quaternion hyperbolic functions also agree with their usual counterparts for real and complex input. This allows us to discuss some important hyperbolic identities and the existence of infinitely many zeros of the quaternion sine and cosine hyperbolic functions, and to solve equations involving hyperbolic functions. A remarkable result of the theory exhibits the deep connection between the hyperbolic and trigonometric functions discussed in the previous chapter. We hope that material presented in this part will make this beautiful topic accessible to the reader.

## Keywords

Trigonometric Function Cosine Function Hyperbolic Function Previous Chapter Complex Exponential Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• João Pedro Morais
• 1
• Svetlin Georgiev
• 2
• Wolfgang Sprößig
• 3
1. 1.CIDMAUniversity of AveiroAveiroPortugal
2. 2.Department of Differential EquationsUniversity of Sofia St Kliment Ohridski Faculty of Mathematics and InformaticsSofiaBulgaria
3. 3.Institut für Angewandte AnalysisTU Bergakademie FreibergFreibergGermany