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Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

Received: 20 March 2016     Accepted: 29 March 2016     Published: 15 April 2016
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Abstract

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.

Published in Applied and Computational Mathematics (Volume 5, Issue 2)
DOI 10.11648/j.acm.20160502.16
Page(s) 73-77
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Julia Set, Mandelbrot Set, Gamma Function, Lanczos Approximation, Complex Functions

References
[1] 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.
[2] 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.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] C. Lanczos, “A precision approximation of the gamma function.” J. Soc. Indust. Appl. Math. Ser. BNumer. Anal., 1964, 1:86-96.
[8] E. R. Scheinerman, Invitation to Dynamical Systems. Dover, U.S.A., 2000, p.1.
[9] 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.
[10] 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.
[11] M. Bourne, Factorials and the Gamma function, 2010: http://intmath.com/blog/mathematics/factorials-and-the-gamma-function-435. Last checked: November 18, 2015.
[12] G. R. Pugh, “An Analysis of the Lanczos Gamma Approximation.” PhD thesis, Department of Mathematics, University of British Columbia, 2004.
Cite This Article
  • APA Style

    Richard Eneojo Amobeda, Terhemen Aboiyar, Solomon Ortwer Adee, Peter Vanenchii Ayoo. (2016). Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation. Applied and Computational Mathematics, 5(2), 73-77. https://doi.org/10.11648/j.acm.20160502.16

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    ACS Style

    Richard Eneojo Amobeda; Terhemen Aboiyar; Solomon Ortwer Adee; Peter Vanenchii Ayoo. Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation. Appl. Comput. Math. 2016, 5(2), 73-77. doi: 10.11648/j.acm.20160502.16

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    AMA Style

    Richard Eneojo Amobeda, Terhemen Aboiyar, Solomon Ortwer Adee, Peter Vanenchii Ayoo. Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation. Appl Comput Math. 2016;5(2):73-77. doi: 10.11648/j.acm.20160502.16

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  • @article{10.11648/j.acm.20160502.16,
      author = {Richard Eneojo Amobeda and Terhemen Aboiyar and Solomon Ortwer Adee and Peter Vanenchii Ayoo},
      title = {Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation},
      journal = {Applied and Computational Mathematics},
      volume = {5},
      number = {2},
      pages = {73-77},
      doi = {10.11648/j.acm.20160502.16},
      url = {https://doi.org/10.11648/j.acm.20160502.16},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160502.16},
      abstract = {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.},
     year = {2016}
    }
    

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    AB  - 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.
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Author Information
  • Department of Mathematics, Kogi State College of Education (Technical), Kabba, Nigeria

  • Department of Mathematics/Statistics/Computer Science, University of Agriculture, Makurdi, Nigeria

  • Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria

  • Department of Mathematics, Federal University Lafia, Nigeria

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