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The argument of the complete elliptic integrals is m=k^2, not k.
After spending a couple of days trying to figure out why I was getting impossible results from some Mma 1.2 MS-DOS 386/7 calculations involving the complete elliptic integrals E(k) and K(k), I discovered that the m given by Mma help as the argument of EllipticE[m] is actually the square of the k that serves in many books as the argument of these functions. Thus, I had to change my EllipticE[k] to EllipticE[k^2] and my EllipticK[k] to EllipticK[k^2]. The heading of Mma help's definition of EllipticK[k] needs to be changed to EllipticK[m], and m needs to be defined in both definitions as k^2. Mma's use of K(m) as the definition of EllipticK[k] should have alerted me more quickly to the problem! It would also have helped if Mma had been able to exhibit a couple of terms of the (Hypergeometric2F1) power series for its E(m) and K(m) around m = 0. Nelson M. Blachman GTE Government Systems Corp. Mountain View, California blachman%gtewd.dnet at bunny.gte.com