# ELLIPTIC INTEGRALS, FUNCTIONS, CURVES AND POLYNOMIALS

### Abstract

The extensive subject of elliptic integrals, functions and curves, being at the junction of analysis, algebra and geometry, has numerous applications in mechanics and physics. Two approaches to the study of elliptic functions have become classical, namely that of Jacobi and that of Weierstrass. Two separate chapters were devoted to these two approaches in the (well-known) course of modern analysis by Whittaker and Watson, without attempting to unite them [1, §§XX, XXII]. Also, two separate chapters are devoted to these two approaches in the latest version (1.0.22 on March 15, 2019) of the NIST Digital Library of Mathematical Functions [2, §§22, 23]. An wide-spread inculcation “explained” that the Weierstrass approach is more suitable for theoretical research, whereas the Jacobi elliptic functions are more common in applications. But, in fact, this dichotomy is artificial, and studying elliptic functions and curves may (and must) be combined in an algebraic approach, establishing a canonical “essential” elliptic function which linear fractional “symmetry” transformations acquire the simplest forms. Although such a natural and fundamental object to be (rightly) called the Galois essential elliptic function, was introduced only recently (already in our millennium), its use has quickly become fruitful, not only and not so much for the effective recovery of known results but also for achieving new calculations that once seemed too cumbersome to pursue. The methodological significance of this natural algebraic approach, which undoubtedly transcends back to the (revolutionary) contribution of Galois, is clearly manifested by its application to several fundamental problems of classical mechanics with the achievement of non-standard, capacious and highly efficient solutions.

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