Integration of forms on the Riemann sphere

Question

Imagine you want to integrate a specific differential form around the equator of the Riemann sphere. This form is such that it is holomorphic at all points above the equator but there is a pole somewhere below the equator. When you stereographically project from the north pole you get the integral around the unit circle of a function which is holomorphic within the unit circle, giving you zero. However when you project from the south pole the singularity lies within the unit circle so you get a non-zero answer. Why don't the two projections agree?