Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation
Applied and Computational Mathematics
Volume 5, Issue 2, April 2016, Pages: 73-77
Received: Mar. 20, 2016; Accepted: Mar. 29, 2016; Published: Apr. 15, 2016
Views 4340      Downloads 174
Richard Eneojo Amobeda, Department of Mathematics, Kogi State College of Education (Technical), Kabba, Nigeria
Terhemen Aboiyar, Department of Mathematics/Statistics/Computer Science, University of Agriculture, Makurdi, Nigeria
Solomon Ortwer Adee, Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria
Peter Vanenchii Ayoo, Department of Mathematics, Federal University Lafia, Nigeria
Article Tools
Follow on us
This work explores the Julia and Mandelbrot sets of the Gamma function by extending the function to the entire complex plane through analytic continuation and functional equations. Various Julia and Mandelbrot sets associated with the Gamma function are generated using the iterative function , with different parameter values. To produce an accurate result using the integral definition of the Gamma function, a large number of terms would have to be added during the numerical integration procedure; this makes computation of Gamma function a very difficult task. To overcome this challenge, the Lanczos approximation of the Gamma function which presents an efficient and easy way to compute algorithms for approximating the Gamma function to an arbitrary precision is used. The resulting images reveal that the fractal (chaotic) behaviour found in elementary functions is also found in the Gamma function. The chaotic nature of the Julia and Mandelbrot sets provides a way of understanding complexity in systems as well as just in shapes.
Julia Set, Mandelbrot Set, Gamma Function, Lanczos Approximation, Complex Functions
To cite this article
Richard Eneojo Amobeda, Terhemen Aboiyar, Solomon Ortwer Adee, Peter Vanenchii Ayoo, Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation, Applied and Computational Mathematics. Vol. 5, No. 2, 2016, pp. 73-77. doi: 10.11648/j.acm.20160502.16
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
J. J. O'Connor and E. F. Robertson, Gaston maurice Julia, 2008. http://www-history.mcs.st-andrews.ac.uk/bibliographies/julia.html. last checked: November 18, 2015.
F. Garcia, A. Fernandez, J. Barrallo, and L. Martin, “Coloring dynamical systems in the complex plane.” The University of the Basque Country, Plaza de O~nati, vol. 2, 2009.
M. Braverman, “Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are Poly Time Computable.” Master's thesis, Graduate Department of Computer Science, University of Toronto, Canada, 2004, 96pp.
S. C. Woon (1998). “Fractals of the Julia and Mandelbrot sets of the Riemann Zeta Function.” Trinity College, University of Cambridge, CB2 ITQ, UK. Accessed at: http://arxiv.org/abs/chao-dyn/9812031v1.
A. Garg, A. Agrawal and A. Negi. “Construction of 3D Mandelbrot Set and Julia Set.” International Journal of Computer Applications, 2014, 85(15): 32-36.
S. Joshi, Y. S. Chauhan, A. Negi. “New Julia and Mandelbrot Sets for Jungck Ishikawa Iterates.” International Journal of Computer Trends and Technology, 2014, 9(5): 209-216.
C. Lanczos, “A precision approximation of the gamma function.” J. Soc. Indust. Appl. Math. Ser. BNumer. Anal., 1964, 1:86-96.
E. R. Scheinerman, Invitation to Dynamical Systems. Dover, U.S.A., 2000, p.1.
L. J. Tingen, The Julia and Mandelbrot sets for the Hurwitz zeta function. Master's thesis, Department of Mathematics and Statistics, University of North Carolina Wilmington, 2009.
K. A. Stroud, and D. J. Booth, Advanced Engineering Mathematics. Palgrave Macmillan, Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N. Y. 10010, 4th edition, 2003.
M. Bourne, Factorials and the Gamma function, 2010: http://intmath.com/blog/mathematics/factorials-and-the-gamma-function-435. Last checked: November 18, 2015.
G. R. Pugh, “An Analysis of the Lanczos Gamma Approximation.” PhD thesis, Department of Mathematics, University of British Columbia, 2004.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186