Elliptic Integrals and Jacobian Elliptic Functions Other methods for computing Dawson’s integral are also known [2,3] . | Elliptic Integrals and Jacobian Elliptic Functions 261 Other methods for computing Dawson s integral are also known 2 3 . CITED REFERENCES AND FURTHER READING Rybicki . 1989 Computers in Physics vol. 3 no. 2 pp. 85-87. 1 Cody . Pociorek . and Thatcher . 1970 Mathematics of Computation vol. 24 pp. 171-178. 2 McCabe . 1974 Mathematics of Computation vol. 28 pp. 811-816. 3 Elliptic Integrals and Jacobian Elliptic Functions Elliptic integrals occur in many applications because any integral of the form y R t S dt where R is a rational function of t and s and s is the square root of a cubic or quartic polynomial in t can be evaluated in terms of elliptic integrals. Standard references 1 describe how to carry out the reduction which was originally done by Legendre. Legendre showed that only three basic elliptic integrals are required. The simplest of these is - Jy I1 dt ai bft a2 b2t 3 bst a4 b t where we have written the quartic s2 in factored form. In standard integral tables 2 one of the limits of integration is always a zero of the quartic while the other limit lies closer than the next zero so that there is no singularity within the interval. To evaluate I1 we simply break the interval y x into subintervals each of which either begins or ends on a singularity. The tables therefore need only distinguish the eight cases in which each of the four zeros ordered according to size appears as the upper or lower limit of integration. In addition when one of the b s in tends to zero the quartic reduces to a cubic with the largest or smallest singularity moving to 1 this leads to eight more cases actually just special cases of the first eight . The sixteen cases in total are then usually tabulated in terms of Legendre s standard elliptic integral of the 1st kind which we will define below. By a change of the variable of integration t the zeros of the quartic are mapped to standard locations on the real axis. Then only two .
