I'm trying to understand why the equal time propagator for a scalar field can sometimes be written in terms of the modified Bessel function $K_1(mr)$ and sometimes in terms of a Yukawa-like potential $\sim \frac{e^{-mr }}{4\pi r} $.
On the one hand, it is regularly argued that the integral that shows up in the Klein-Gordon equal time propagator can be written in terms of the modified Bessel function $K_1$: $$ D(\vec x,\vec x') \equiv \langle \vec x' | \vec x\rangle = \int \frac{ \mathrm{d }k^3 }{(2\pi)^3 2\omega_{k} } {\mathrm{e }}^{- i\vec k \cdot (\vec x ' -\vec x)} =\frac{m}{4\pi^2 r} K_1(mr),$$ where $r \equiv |\vec x - \vec x'|$.
On the other hand, the Green's function of the Klein-Gordon equation for a static field configuration $\phi$ reads $$G(\vec x,\vec x') = \int \frac{dk^3}{(2\pi)^3} \; \frac{e^{-i \vec k \cdot (\vec x - \vec x')}}{-k^2 + m^2}= \frac{e^{-mr }}{4\pi r} $$ where again $r \equiv | \vec x - \vec x'| $. (The integral is solved explicitly in Zee's book on page 29.)
Shouldn't the equal time propagator be equal to the Green's function for a static field configuration? If yes, why do we get such as simple solution (a Yukawa potential) in the second case and a much more complicated solution (a modified Bessel function) in the former case?
A useful hint is probably that the propagator for spacelike separations decays approximately like $e^{-mr }$: $$ D(\vec x,\vec x') \equiv \langle \vec x' | \vec x\rangle = \int \frac{ \mathrm{d }k^3 }{(2\pi)^3 2\omega_{k} } {\mathrm{e }}^{- i\vec k \cdot (\vec x ' -\vec x)} \sim e^{-mr }$$ (This is demonstrated at page 18 here, for example, or on page 27 in Peskin & Schröder's book)
