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 $p dp/dz - p = z$ satisfying $p = -z + O$ 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 $p$ which is used to give the connection formula for the problem $r^2 f^{''} + f = f^3$, $f = 0, f = 1$.