Volume 20, pp. 198-211, 2005.

Recursive computation of certain integrals of elliptic type

P. G. Novario

Abstract

An algorithm for the numerical calculation of the integral function
$N_n(x)={\int }_0^{\pi /2}\frac{{\cos }^{2n}(\mathrm{\Phi })}{\sqrt{1-x\cdot {\sin }^2(\mathrm{\Phi })}}\cdot d\mathrm{\Phi }(0\le x<1;n=0,1,2,\dots )$,
distinguished solution of the second-order difference equation
$(2n+1)\cdot x\cdot N_{n+1}(x)+2n\cdot (1-2x)\cdot N_n(x)=(2n-1)\cdot (1-x)\cdot N_{n-1}(x)(n=1,2,\dots )$,
that uses the recurrence relation and its related continued fraction expansion, is described and discussed. The numerical efficiency of the algorithm is analysed for various x values of the interval ($0\le x<1$). A twelve digits tabulation of $N_n(x)$ for $n=1(1)20$ and $x=0(0.02)1$ is presented as example of the algorithm utilization.

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Key words

recurrence relations, elliptic integrals, continued fractions

AMS subject classifications

65Q05, 33E05, 11A55