This note looks at the relation between inverse poles and nearby nodes, working the simplest example, an inverse rational function defioed by an equation
     (w-P)z = f(w)
where f is a polynomial, and P is the (only) inverse pole. Leaving aside the trivial cases where f is constant or linear a branch-point may be finite or infinite. In the first case the two equations for a node and branch-point pair W,Z are
     (W-P)Z = f(W) and Z = f'(W).
Eliminating Z gives
     (W-P)f'(W) = f(W).
Expressing this as a power series in W-P gives after a little calculation
     0 = f(P) - (W-P)²f''(P)/2 - (W-P)³f'''(P)/3  ...-(W-P)ⁿf⁽ⁿ⁾(P)(n-1)/(n!)...
Now assume that f has parameters which allow W to be arbitrarily near P. This applied gives the approximation
f(P) ≅ (W-P)²f''(P)/2
indicating two such nodes on opposide sides of P. Let the two corresponding branch-points be joined by a suitable finite cut. This is imaged by a boundary which bounds the range of a branch which has P interior to it.