Elliptic_Functions

Elliptic Functions The terminology for elliptic integrals and functions has changed during their investigation. What were originally called elliptic functions are now called elliptic integrals and the term elliptic functions reserved for a different idea. We will therefore use modern terminology throughout this article to avoid confusion. It is important to understand how mathematicians thought differently at different periods. Early algebraists had to prove their formulas by geometry. Similarly early workers with integration considered their problems solved if they could relate an integral to a geometric object. Many integrals arose from attempts to solve mechanical problems. For example the period of a simple pendulum was found to be related to an integral which expressed arc length but no form could be found in terms of 'simple' functions. The same was true for the deflection of a thin elastic bar. The study of elliptical integrals can be said to start in 1655 when Wallis began to study the arc length of an ellipse. In fact he considered the arc lengths of various cycloids and related these arc lengths to that of the ellipse. Both Wallis and Newton published an infinite series expansion for the arc length of the ellipse. At this point we integral, elliptic, arc, length, functions, integrals, lemniscate, t2, jacob, form, example, ellipse, dt/1, curve, considered, bernoulli, wallis, terms, terminology, t4, study, simple, related, px, problems, lengths, important, given, geometric, general, function, found, fact, expressed, elastic