## Abstract

The integral equation $\{F(z)\}^{4}$ = $-\frac{1}{2}z^{-\frac{1}{2}}$ $\int_{0}^{z}$ $\frac{F^{\prime}(u)\text{d}u}{(z^{\frac{3}{2}}-u^{\frac{3}{2}})^{\frac{1}{3}}}$, $F(0)=1$, arose in a problem studied by Lighthill (1950). A series expansion for small $z$, $F(z)=\underset n=0\to{\overset \infty \to{\Sigma}}a_{n}z^{n}$, is shown, by repeated extrapolation, to have a radius of convergence $z_{0}$ of about 0.106 058 0 to within 1 or 2 units of the last figure. An Euler transformation $\xi $ = $z(z+z_{0})^{-1}$ is then applied, yielding $F(\xi)=\underset n=0\to{\overset \infty \to{\Sigma}}b_{n}\xi ^{n}$, which converges much better than the previous series. Likewise Lighthill derived an asymptotic expansion for large $z$, and the first 6 terms are accurately calculated here. When written in terms of $\xi $, the expansion yields $F(\xi)\approx \underset n=1\to{\Sigma}B_{n}(1-\xi)^{\frac{1}{4}n}$. When the first six terms are expanded in powers of $\xi $, and substracted from the series for $F(\xi)$, we find that $F(\xi)=\underset n=1\to{\overset 6\to{\Sigma}}B_{n}(1-\xi)^{\frac{1}{4}n}+\underset n=0\to{\overset \infty \to{\Sigma}}c_{n}\xi ^{n}$, where several $c_{n}$ and the six $B_{n}$ are given. The convergence of the series is now such that when $z$ = 1, i.e. $\xi $ = (1 + $z_{0})^{-1}\approx $ 0.904 $F(\xi)$ may be calculated to better than 9 decimal places. Even when $\xi $ = 1, where the convergence is least good, $F(\xi)$ is given to better than 6 decimal places. In each of these calculations the convergence is accelerated by repeated Aitken extrapolation.