Fractals as Julia Sets for a New Complex Function via a Viscosity Approximation Type Iterative Methods
In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mrow><mo...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/850 |
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| Summary: | In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><msup><mi>e</mi><msup><mi>z</mi><mi>n</mi></msup></msup><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>β</mi><msup><mi>z</mi><mi>m</mi></msup></mfrac></mstyle><mo>+</mo><mo form="prefix">log</mo><msup><mi>c</mi><mi>t</mi></msup><mo>,</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><msup><mi>e</mi><msup><mi>z</mi><mi>n</mi></msup></msup><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>β</mi><msup><mi>z</mi><mi>m</mi></msup></mfrac></mstyle><mo>+</mo><mi>γ</mi></mrow></semantics></math></inline-formula> (which are analytic except at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>,</mo></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mi>t</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> by employing a viscosity approximation-type iterative method. We employ the proposed iterative method to establish an escape criterion for visualizing Julia sets. We provide graphical illustrations of Julia sets that emphasize their sensitivity to different iteration parameters. We present graphical illustrations of Julia sets; the color, size, and shape of the images change with variations in the iteration parameters. With fixed input parameters, we observe the intriguing behavior of Julia sets for different values of <i>n</i> and <i>m</i>. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry. |
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| ISSN: | 2075-1680 |