Novel escape criteria for complex-valued hyperbolic functions through a fixed point iteration method

Tunçar Şahan; Yunus Atalan; Tunçar Şahan; Yunus Atalan

doi:10.3934/math.2025071

AIMS Mathematics

2025, Volume 10, Issue 1: 1529-1554. doi: 10.3934/math.2025071

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Research article

Novel escape criteria for complex-valued hyperbolic functions through a fixed point iteration method

Tunçar Şahan ,
Yunus Atalan ^,

Department of Mathematics, Aksaray University, Aksaray, 68100, Türkiye

Received: 28 November 2024 Revised: 08 January 2025 Accepted: 15 January 2025 Published: 23 January 2025
MSC : 47H10, 28A80

This study presented an efficient fixed-point iteration method for deriving novel escape criteria for hyperbolic sine and hyperbolic cosine functions of varying degrees. The method contributes to obtaining more accurate and effective escape criteria, thereby enhancing the mathematical understanding and computational analysis of these functions. Additionally, using the derived criteria, the iteration method was employed to generate visually appealing fractals for Julia and Mandelbrot sets, demonstrating significant advantages in computational speed and practical utility. The method's effective performance in producing complex and aesthetically satisfying fractal structures highlights its efficiency and applicability in fractal generation.
- fixed-point,
- iteration method,
- escape criterion,
- Julia set,
- Mandelbrot set
Citation: Tunçar Şahan, Yunus Atalan. Novel escape criteria for complex-valued hyperbolic functions through a fixed point iteration method[J]. AIMS Mathematics, 2025, 10(1): 1529-1554. doi: 10.3934/math.2025071

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Abstract

This study presented an efficient fixed-point iteration method for deriving novel escape criteria for hyperbolic sine and hyperbolic cosine functions of varying degrees. The method contributes to obtaining more accurate and effective escape criteria, thereby enhancing the mathematical understanding and computational analysis of these functions. Additionally, using the derived criteria, the iteration method was employed to generate visually appealing fractals for Julia and Mandelbrot sets, demonstrating significant advantages in computational speed and practical utility. The method's effective performance in producing complex and aesthetically satisfying fractal structures highlights its efficiency and applicability in fractal generation.

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