Boundary Value Problem for $r^2{d}^{2}f/d{r}^2+f=f^3$ (II): Connection Formula

Abstract

In this paper, we study the analytic expansion in a small neighborhood of 0 in the complex plane for the solution to the equation $pdp/dz-p=z(z-1)(z-2)$ satisfying $p(z)=-z+O(z^2)$ as $z\to 0$. We show that the expansion is valid for $|z|\le s_0$, where $s_0>1$. Then we get an explicit formula for $p(1)$ which is used to give the connection formula for the problem $r^2{f}^{{}^{\prime \prime }}+f=f^3$, $f(1)=0,f(\infty )=1$.

12 pages