Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

Richard Eneojo Amobeda; Terhemen Aboiyar; Solomon Ortwer Adee; Peter Vanenchii Ayoo

doi:doi:10.11648/j.acm.20160502.16

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Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

Richard Eneojo Amobeda, Terhemen Aboiyar, Solomon Ortwer Adee, Peter Vanenchii Ayoo

Published in Applied and Computational Mathematics (Volume 5, Issue 2)

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, 4^th 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.

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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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TY  - JOUR
T1  - Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation
AU  - Richard Eneojo Amobeda
AU  - Terhemen Aboiyar
AU  - Solomon Ortwer Adee
AU  - Peter Vanenchii Ayoo
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N1  - https://doi.org/10.11648/j.acm.20160502.16
DO  - 10.11648/j.acm.20160502.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
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EP  - 77
PB  - Science Publishing Group
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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.
VL  - 5
IS  - 2
ER  -

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Author Information

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

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation
AU  - Richard Eneojo Amobeda
AU  - Terhemen Aboiyar
AU  - Solomon Ortwer Adee
AU  - Peter Vanenchii Ayoo
Y1  - 2016/04/15
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160502.16
DO  - 10.11648/j.acm.20160502.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 73
EP  - 77
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160502.16
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.
VL  - 5
IS  - 2
ER  -

Copy | Download