THE COALESCENCE LIMIT OF THE 2ND PAINLEVE EQUATION

In this paper, we study a well-known asymptotic limit in which the sec ond Painleve equation (P-II) becomes the first Painleve equation (P-I) . The limit preserves the Painleve property (i.e., that all movable si ngularities of all solutions are poles). Indeed it has been commonly a ccepted that the movable simple poles of opposite residue of the gener ic solution of P-II must coalesce in the limit to become movable doubl e poles of the solutions of P-I, even though the limit naively carried out on the Laurent expansion of any solution of P-II makes no sense, Here we show rigorously that a coalescence of poles occurs. Moreover w e show that locally all analytic solutions of P-I arise as limits of s olutions of P-II.