Boundary Value Problem for $r^2 d^2 f/dr^2 + f = f^3$ (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 $p dp/dz - p = z(z-1)(z-2)$ 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 $r^2 f^{''} + f = f^3$, $f(1) = 0, f(\infty) = 1$.