Abstract
In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.
Highlights
Elliptic functions were first discovered by Niels Henrik Abel [2] as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives
Jacobi’s elliptic functions have found numerous applications in physics and were used by Jacobi to prove some results in elementary number theory. e solution of the quintic algebraic equation was provided by Hermite
Any entire algebraic equation of nth degree may be expressed through elliptic theta functions [3]. e elliptic functions have a variety of applications in science and technology. ese functions are the main tool for solving nonlinear differential equations
Summary
Elliptic functions were first discovered by Niels Henrik Abel [2] as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Ese functions are the main tool for solving nonlinear differential equations. These functions, as well as elliptic integrals, are difficult to evaluate. E main Jacobian elliptic functions may be expressed by means of trigonometric series. Truncating these series, we get approximate analytic expressions in terms of the elementary trigonometric function. Is function allows to solve the following nonlinear ode called Duffing–Helmholtz equation: x€ + n + px + qx2 + rx[3 0], x(0) x0,. Many identities related to elliptic functions and elliptic integrals may be found in [1, 4] Taking into account these difficulties, there is a need to obtain elementary and reasonable accurate approximate analytical expressions for these functions and their inverses.
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