On Julia and Mandelbrot sets of gamma and beta functions for fractal images

Abstract

This work explores the Julia and Mandelbrot sets of the gamma and beta functions by 
using Lanczos approximation procedures for fractal images. Various Julia and Mandelbrot 
sets associated with the gamma function are generated using the iterative function, fλ
 
(z), with varoius λ-values. The Lanczos approximation of the gamma function presents an 
efficient and easy algorithm to compute an approximate solution of the iterative gamma 
function to an arbitrary precision. The resulting images reveal that the fractal (chaotic) 
behavior found in elementary functions is also found in the gamma functions. The chaotic 
nature of the Julia and Mandelbrot sets provides an understanding of complexity in sys
tems as well as in shapes. The relationship between the gamma and beta functions was 
exploited to obtain the Lanczos approximation for beta function.