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Asymptotic expansion of the first elliptic integral
Ist Teil von
SIAM journal on mathematical analysis, 1985-09, Vol.16 (5), p.1072-1092
Ort / Verlag
Philadelphia, PA: Society for Industrial and Applied Mathematics
Erscheinungsjahr
1985
Link zum Volltext
Quelle
Alma/SFX Local Collection
Beschreibungen/Notizen
Asymptotic formulas with error bounds are obtained for the first elliptic integral near its logarithmic singularity. It is convenient to start from the more general problem of expanding the integral over the positive $t$-axis of $[(t + x)(t + y)(t + z)(t + w)]^{{{ - 1} / 2}} $. The method of Mellin transforms gives an asymptotic expansion that converges uniformly if ${{0 < \max \{ x,y\} } / {\min \{ z,w\} \leqq r < 1}}$. Each term of the series contains a Legendre polynomial and the derivative of a Legendre function with respect to its degree; this derivative involves nothing worse than a logarithm. A simple bound for the relative error of the $N$th partial sum is obtained from Wong's formula for the remainder, aided by Chebyshev's integral inequality. Error bounds are given also for more accurate asymptotic formulas containing a complete elliptic integral. Formulas for the standard elliptic integral of the first kind are obtained in the case $w = \infty $. The method of Mellin transforms and Wong's formula are discussed in an appendix.