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$$\Delta _ { F } ( x - y ) = \int \frac { d ^ { 4 } p } { ( 2 \pi ) ^ { 4 } } \frac { i } { p ^ { 2 } - m ^ { 2 } } e ^ { - i p \cdot ( x - y ) },$$

there are poles on the real axis. I have seen several different prescriptions for avoiding the poles using contour integration such as rotating the integral into the complex plane, traveling around the poles in tiny semicircular paths and adding an infinitesimal complex term in the denominator.

$$\Delta _ { F } ( x - y ) = \int \frac { d ^ { 4 } p } { ( 2 \pi ) ^ { 4 } } \frac { i e ^ { - i p \cdot ( x - y ) } } { p ^ { 2 } - m ^ { 2 } + i \epsilon }$$

How can we say that these modified integrals are equal to the Feynman Propagator? Didn't we fundamentally change it to avoid the poles? We got rid of an infinity after all.