## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2016 (2016), Article ID 3249439, 18 pages.

### $p$-Trigonometric and $p$-Hyperbolic Functions in Complex Domain

#### Abstract

We study extension of $p$-trigonometric functions ${\mathrm{s}\mathrm{i}\mathrm{n}}_{p}$ and ${\mathrm{c}\mathrm{o}\mathrm{s}}_{p}$ and of $p$-hyperbolic functions ${\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_{p}$ and ${\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}_{p}$ to complex domain. Our aim is to answer the question under what conditions on $p$ these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, $\mathrm{s}\mathrm{i}\mathrm{n}(z)=-i\xb7\mathrm{sinh}\left(i\xb7z\right)$. In particular, we prove in the paper that for $p=\mathrm{6,10,14},\dots $ the $p$-trigonometric and $p$-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series for $p$-trigonometric and $p$-hyperbolic functions.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2016 (2016), Article ID 3249439, 18 pages.

**Dates**

Received: 28 July 2015

Accepted: 1 December 2015

First available in Project Euclid: 19 May 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1463662622

**Digital Object Identifier**

doi:10.1155/2016/3249439

**Mathematical Reviews number (MathSciNet)**

MR3498048

**Zentralblatt MATH identifier**

1349.33018

#### Citation

Girg, Petr; Kotrla, Lukáš. $p$ -Trigonometric and $p$ -Hyperbolic Functions in Complex Domain. Abstr. Appl. Anal. 2016 (2016), Article ID 3249439, 18 pages. doi:10.1155/2016/3249439. https://projecteuclid.org/euclid.aaa/1463662622