# On deriving the standard form of the Klein-Gordon propagator

I'm trying to make sense of the derivation of the Klein-Gordon propagator in Peskin and Schroeder using contour integration. It seems the main step in the argument is that ## e^{-i p^0(x^0-y^0)} ## tends to zero (in the ##r\rightarrow\infty## limit) along a semicircular contour below (resp. above) the real line for ##x^0 > y^0## (resp. ## x^0< y^0 ##). It is this that I'm having trouble making sense of. If we consider $$\int_{C_{below}} \frac{dp^0}{2\pi}\frac{e^{ip(x--y)}}{i(p^2-m^2)} = \int_{0}^\pi \frac{d\theta}{2\pi}(-ire^{i\theta})\frac{e^{-i[re^{i\theta}(x^0-y^0) - \vec{p}\cdot(\vec{x}-\vec{y})]}}{i((re^{i\theta})^2 - \vec{p}^2 - m^2)},$$ the integrand seems to tend to zero as ## r\rightarrow\infty ## regardless of whether ##x^0 > y^0## or ##y^0 > x^0## since the exponential is bounded. So I guess my question is what stops us from choosing this same contour for the ##y^0 > x^0## case, which would give a nonzero result?