Inverse Elliptic Functions and Legendre Polynomials

Power series for inverse Jacobian elliptic functions

“…If x and y are real with opposite signs, all quantities in (2.7) are real, but γ and the variable of P n are pure imaginary. This unnecessary complication occurs, for example, in arcsd if 0 < k 2 < 1; it is dealt with in [Ke,(5),(6)] by introducing a second polynomial S n that differs from P n by changing minus signs to plus signs. The polynomial R n (x, y) suffices for both signs of xy, and its symmetry was important in showing that the four series in each of three subsets have the same coefficients.…”

“…(Incidentally, in [Ke,(2)] a function R n (x) is defined that can be shown to be R n (1, x 2 ) in the notation used here.) Why, then, use R n instead of P n in Theorem 1?…”

“…The equations for arccn given by [Ke,(6)] and [Wo] follow from the power series for arcsd and the relation arccn(z, k) = K(k) − arcsd(z/k ) [La,(3.2.18)]. …”

“…Only partial information about power series for inverse Jacobian elliptic functions seems to be available if we ask for series in which all coefficients are algebraic functions of the modulus k. If z = sn (u, k), the inverse function is u = arcsn(z, k), which was expanded in power series by Kelisky [Ke,(4)]:…”

Some Series and Bounds for Incomplete Elliptic Integrals

“…As an example, we shall expand the three elliptic integrals in powers of sin ¢ and show that the coefficients are Legendre polynomials. The first of these expansions was found recently by Kelisky [8] by a quite different method; the other two appear to be new:…”