Elliptic Functions

are generalizations of the circular functions sine , cosine , . If IMG:3349192321007894546_ellfunc1.png:Equation we have x = sin u. Similarly, if IMG:3349192321007894546_ellfunc2.png:Equation we may write x = sn u , √(1 - x 2 ) = cn u , √(1 - k 2 x 2 ) = dn u . These are Jacobi's elliptic functions. They obviously reduce to sin u , cos u , 1, when k is 0. The circular functions have the period 2 π ; the elliptic functions are doubly periodic, having both a real and a pure imaginary period when k 2 is real and le than 1. Like the functions sine and cosine , the elliptic functions have addition theorems , e.g. | sn ( u + v ) = | sn u cn v dn v + sn v cn u dn u |. | | 1 - k 2 sn 2 u sn 2 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 .