An Elementary Introduction to Elliptic Functions Based on the Theory of Nutation

Abstract

Introduction. Elliptic functions are a classical branch of mathematics, and recur in many problems of physics and engineering -ither directly, or as a skeleton of more complicated problems. Yet it cannot be denied that they are known nowhere as widely as they deserve. In many problems when the elliptic functions could advantageously be used, or at least inspected as a possibly useful first approximation, it is usually possible to find an even simpler (and not always adequate) skeleton of the theory in trigonometry. Perhaps this is so because the elliptic functions are traditionally presented not as a refinement on trigonometry, but as a subject in the theory of functions of a complex variable; trigonometry, however, is never presented-and in fact, seldom reviewed-as such a subject. Indeed, there is no need to encumber a presentation of the circular functions to a beginner by a simultaneous presentation of the hyperbolic functions and of all relations between these two sets of functions. Nor would such a presentation be particularly instructive: for the poles of sin and cos are at an imaginary infinity (and hence cannot be circled), while those of tan, cot, sec and cosec lie simply on the real axis. Moreover, the elliptic functions are traditionally presented not as solutions of differential equations, but as inversions of certain integrals. This is as though one introduced trigonometry by defining the arcsin as an incomplete circular integral of the first kind, or worse, as though w were introduced as twice the complete circular integral. The tradition evolved in the pre-computer era, when the numerical solution of differential equations was considered just another theoretical method, and not a very practical method at that, of specifying a new function. The solution was considered completed only when reduced to quadrature or series and interpolation; and it was therefore considered natural that much algebraic work had to be done before one resorted to computations. In contrast, now, when torrents of new functions are being generated on machines and the bottlenecks lie in the analytical work, the definition of a function by its differential equations is not only the more convenient, but appears to be a basic method. A student naturally expects that the elliptic functions are a generalization of the circular functions, and are therefore more complicated; but he hopes that the new labor would be, in proportion, no more than, say, going from a circle to an ellipse, or from plane trigonometry to the spherical trigonometry. Instead, he finds an entirely new approach, and a new theory, which only retrospectively reduces to trigonometry-and a rather advanced trigonometry at that-as the poles of the elliptic analogues of sin and cos recede to imaginary infinity. The forward step (from circular to elliptic functions) requires turning this infinity into a large finite number. It would have been easier if this step involved an appearance of something small which was zero before; as a matter of fact, it is the latter which is provided-in various disguises-by his physical problems.