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The influence one event, process, or state, has on another event, process, or state, whereby the latter is at least partly dependent on the former.
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Bessel function representation of spacelike KG propagator
This can be seen by partial integration
$$\frac{\partial}{\partial \rho}\sqrt{\rho^2-m^2}=\frac{\rho}{\sqrt{\rho^2-m^2}}$$
OP edit: More explicitly, we use this to write $(3)$ as
\begin{align}
D( …