New series for the cosine lemniscate function and the polynomialization of the lemniscate integral

摘要

We discuss the numerical computation of the cosine lemniscate function and its inverse, the lemniscate integral. These were previously studied by Bernoulli, Euler, Gauss, Abel, Jacobi and Ramanujan. We review general elliptic formulas for this special case and provide some novelties. We show that a Fourier series by Ramanujan converges twice as fast as the standard elliptic cosine Fourier series specialized to this case. The so-called imbricate series, however, converges geometrically fast over the entire complex plane. We derive two new expansions. The rational-plus-Fourier series converges much faster than Ramanujan’s: for real z: each term is asymptotically 12,400 times smaller than its immediate predecessor: coslem(z)=4B{q(1−q)cos(Bz)/[(1+q)2−4qcos2(Bz)]+∑n=1∞qn−1/2{1/(1+q2n−1)−1}cos((2n−1)Bz)} where q=exp(−π) is the elliptic nome, K≈1.85… is the complete elliptic integral of the first kind for a modulus m=1/2 and B=π/(K2). The rational imbricate series is uniformly valid over the complex plane, but converges twice as fast as the sech-imbricate series: coslem(z)=4Bq(1−q)∑j=−∞∞q(1−q)cos(Bz)/{(1+q)2−4qcos2(B[z−jPi])} where P=(4/2)K is the period in both the real and imaginary directions.We devise a new approximation for the lemniscate integral for real argument as the arccosine of a Chebyshev series and show that 17 terms yield about 15 digits of accuracy. For complex argument, we show that the lemniscate integral can be found to near machine precision (assumed as sixteen decimal digits) by computing the roots of a polynomial of degree thirteen. Alternatively, Newton’s iteration converges in three iterations with an initialization, accurate to four decimal places, that is the chosen root of a cubic equation.