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 W(z)=αezn+βzm+logct, and T(z)=αezn+βzm+γ (which are analytic except at z=0) where n≥2, m,n∈N, α,β,γ∈C,c∈C∖{0} and t∈R,t≥1, 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 n and m. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry.