Bounds for Weierstrass Elliptic Function and Jacobi Elliptic Integrals of First and Second Kinds

  • Stephen Ehidiamhen Uwamusi Department of Mathematics, Faculty of Physical Sciences, University of Benin, Benin City, Edo State, Nigeria
Keywords: complex meromorphic function, Weierstrass elliptic function, Eisenstein series, Jacobi elliptic integrals, Konrod-Gauss quadrature method, Runge-Kutta method


The Weierstrass elliptic function is presented in connection with the Jacobi elliptic integrals of first and second kinds leading to comparing coefficients appearing in the Laurent series expansion with those of Eisenstein series for the cubic polynomial in the meromorphic Weierstrass function.

It is unified in the formulation the Weierstrass elliptic function with Jacobi elliptic integral by considering motion of a unit mass particle in a cubic potential in terms of bounded and unbounded velocities and the time of flight with imaginary part in the complex function playing a major role. Numerical tools box used are the Konrad-Gauss quadrature and Runge-Kutta fourth order method.


Abramowitz, M., & Stegun, I. A. (Eds.). (1992). Handbook of mathematical functions with formulas, graphs and mathematical tables (Reprint of 1972 edition). Dover Publications Inc.

Allen, M. A. (2018). Elliptic integrals and the Jacobi elliptic functions. Seminar delivered in Physics Department, Mahidol University, Bangkok.

Alzer, H., & Richards, K. C. (2020). A concavity property of the complete elliptic integral of the first kind. Integral Transform Spec. Funct., 31(9), 758-768.

Alzer, H., & Qiu, S. L. (2004). Monotonicity theorems and inequalities for the complete elliptic integrals. Journal of Computational and Applied Mathematics, 172(2), 289-312.

Anderson, G. D., Qiu, S. L., & Vamanamurthy, M. K. (1998). Elliptic integrals, with applications. Constructive Approximation, 14(2), 195-207.

Brizard, A. J. (2015). Notes on the Weierstrass elliptic function. arXiv:1510.07818V1 [].

Chu, Y., Wang, M., & Qiu, Y. (2011). On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function. Abstract and Applied Analysis, 2011, Article ID 697547, 7 pp.

Clemons, J. J. (2010). Dynamical properties of Weiertstrass elliptic functions on square lattice. PhD Thesis, Department of Mathematics, University of North Carolina at Chapel Hill, USA.

Coquereaux, R., Grossmann, A., & Lautrup, B. E. (1990). Iterative method for calculation of the Weierstrass elliptic function. IMA Journal of Numerical Analysis, 10, 119-128.

Ellbeck, J. C., England, M., & Onishi, Y. (2021). Some new addition formulae for Weierstrass elliptic function. Proceedings of the Royal Society, 1-14.

Ibrahim, J. (2012). Applications of the arithmetic theory of elliptic curves. PhD Thesis, Department of Mathematics, Universidad de Guanajuato.

Izzo, D. (2016). On the astrodynamics application of Weierstrass elliptic and related functions. AAS/AIAA Space flight mechanics meetings, Napa, CA in February, 14-16.

Johanson, F. (2016). Computing hypergeometric functions rigorously. Ha-01336266v2.

Johannessen, K. (2018). The solution to the differential equation with linear damping describing the physical systems governed by a cubic energy potential. Retrieved from

Krakiwsky, E. J., & Thomson, D. B. (1995). Geodetic position computations. Department of Geodesy and Geomatics Engineering, University of New Brunswick, Canada.

Koss, L. (2014). Examples of parameterized families of elliptic functions with empty Fatou sets. New York Journal of Mathematics, 20, 607-625.

Lawden, D. F. (1989). Elliptic functions and applications. Springer Science + Business Media LLC.

Okeke, E. O. (1990). Theory of nonlinear water waves. M.Sc. Lecture Notes, Department of Mathematics, University of Benin, Benin City, Nigeria.

Pastras, G. (2017). Four lectures on Weierstrass elliptic function and applications in classical and quantum mechanics. NCSR “Demokritos’’, Institue of Nuclear and Particle Physics, Parakevi, Attiki, Greece.

Qiu, S. L., Vamanamurthy, M. K., & Vuorinen, M. (1998). Some inequalities for the growth of elliptic integrals. SIAM J. Math. Anal., 29(1), 1-6.

Schlosser, M. J. (2008). A Taylor expansion theorem for an elliptic extension of the Askey-Wilson operator. arXiv:0803.2329v1[math.CA].

Taylor, M. (2018). Elliptic functions. Sections 30-34 of Introduction to complex analysis. Retrieved from

Uwamusi, S. E. (2021). The gamma and beta matrix function and other applications. Unilag Journal of Mathematics and Applications, 1(2), 186-207

Wang, M. K., Chu, H. H., Li, Y. M., & Chu, Y. M. (2020). Answers to three conjectures on convexity of three functions of the first kind. Appl. Anal. Discrete Math., 14, 255-271.

Whittaker, E. T., & Watson, G. N. (1963). A course of modern analysis. Cambridge University Press.

Yang, Z., & Tian, J. (2017). Convexity and monotonicity for the elliptic integrals of the first kind and applications. arXiv:1705.05703v1 [Math.CA].

How to Cite
Uwamusi, S. E. (2024). Bounds for Weierstrass Elliptic Function and Jacobi Elliptic Integrals of First and Second Kinds. Earthline Journal of Mathematical Sciences, 14(3), 535-564.