 The Theory of Elliptic Functions Based on Elementary Analysis

DOI：10.3770/j.issn:1000-341X.1981.03.016

 作者 单位 莫绍揆 南京大学数学系

The prevailing approach of the theory of elliptic functions, whether classic (as the inverse of elliptic integrals of the first kind) or following Weierstrass (as the double periodic meromorphic functions), is based on some deep results of the theory of functions of complexe variables, and hence is too difficult to the beginners. However, on the other hand, it is quite well known that the theory of elliptic functions is essentially very elementary, accessible to the precalculus students,In the present paper, we base the theory of elliptic functions on the phenomena of simple pendulum. From this we deduce successively the fundamental relations, the periodicity, the derivatives, and finally, the addition theorems of the elliptic functions. And then it would be easy to develop the whole theory. Besides, in order to make the theory really simple and easy, we must reform it in many respects. For example, we need choose suitable basic functions, standardize the periods and the leading coefficients of various functions. It is obvious uthat we should take 2K and 2iK′ as periods. Besides, we should take as basic the following functions with suitable leading coefficients: the twe1ve Jacobian and Glaisher's functions and the functioils P,ξ,σ, D, Z,θ,п, each with one of the four subscripts s, c, d, n.