## Recursive computation of certain integrals of elliptic type

P. G. Novario

### Abstract

An algorithm for the numerical calculation of the integral function
$N_{n}(x)=\int^{\pi/2}_{0}\frac{\cos^{2n}(\Phi)}{\sqrt{1-x\cdot\sin^{2}(\Phi)}}\cdot d\Phi\,\,\,\,(0\leq x < 1 ; \;n=0,1,2,\ldots)$,
distinguished solution of the second-order difference equation
$(2n+1)\cdot x\cdot N_{n+1}(x) + 2n\cdot (1-2x)\cdot N_{n}(x)=(2n-1)\cdot (1-x)\cdot N_{n-1}(x)\;\;(n=1, 2, \ldots)$,
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 ($0\leq x < 1$). 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

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