ELLIPTIC FUNCTIONS

are gener| a liz a tions of the circular functions sine, cosine, . dx x way ey iles a we have x = sin wu. Similarly, if Uu= POE en aie eB o V (1 — x) (1 — k*x?) » we may write s = snu, J/1 — a? = cnu, 1 — kx? = dnu. These are Jacobi’s elliptic functions. They = sin—z, obviously reduce to sin u, cos u,1, when kis0. The circular functions have the period 27; the elliptic functions are doubly periodic, having both a real and a pure imaginary period when k? is real and le than 1. Like the functions sine and cosine, the elliptic funetions have addition theorems, e.g. snucnvdnv + snvcnudnu 1 — k'sn*u sn®v Another method and notation has been introduced by We i ers tra , and is now much used. The functions are needed for the solution of many physical problems, such as those of the motion of a top and of a pendulum. —BIBLIOGRAPHY: A. G. Green hill, Elliptic Functions; Appell and Lacour, Fonctions Elliptiques; Whittaker and Watson, Modern Analysis.