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## Main Question or Discussion Point

I'm trying to derive the x-space result for the Green's function for the Klein-Gordon equation, but my complex analysis skills seems to be insufficient. The result should be:

\begin{eqnarray}

G_F(x,x') = \lim_{\epsilon \rightarrow 0} \frac{1}{(2 \pi)^4} \int d^4k\frac{e^{-ik\cdot(x-x')}}{k^2-\frac{m^2c^2}{\hbar^2}+i\epsilon}

= \left(-\frac{1}{4\pi} \delta (s^2) + \frac{mc^2}{8 \pi \hbar s} H_1^{(1)} \left(\frac{mc^2}{\hbar} s \right) \right) \:\Theta (s)

- \frac{m}{4\pi^2 s} K_1 \left(i\frac{mc^2}{\hbar} s \right) \:\Theta(-s))\\

\end{eqnarray}

with [itex]s^2 = c^2(t-t')^2-(\vec{x}-\vec{x}')^2[/itex], the source point [itex]x'[/itex], [itex]\Theta(s)[/itex] the Heaviside stepfunction, [itex]H_1^{(1)}(x)[/itex] the Hankel function of the first kind, and [itex]K_1(x)[/itex] the modified Bessel function of the second kind.

I changed to spherical polar coordinates and did the [itex]\phi[/itex] and [itex]\theta[/itex] integrals first. From there I've tried several approaches, but I always end up with a complex integral of a multivalued function:

\begin{equation}

\int \frac{f(z)}{\sqrt{z^2 \pm a^2}} dz

\end{equation}

From residues I get zero when [itex]f(z)[/itex] is an exponential function, but that can't be right?

Both my quantum field theory and many-particle books work in k-space and I haven't found the derivation here.

Would someone be kind enough to guide me throught the correct procedure for the different steps of the Fourier transform of the Feynman propagator?

Thanks for your time!

\begin{eqnarray}

G_F(x,x') = \lim_{\epsilon \rightarrow 0} \frac{1}{(2 \pi)^4} \int d^4k\frac{e^{-ik\cdot(x-x')}}{k^2-\frac{m^2c^2}{\hbar^2}+i\epsilon}

= \left(-\frac{1}{4\pi} \delta (s^2) + \frac{mc^2}{8 \pi \hbar s} H_1^{(1)} \left(\frac{mc^2}{\hbar} s \right) \right) \:\Theta (s)

- \frac{m}{4\pi^2 s} K_1 \left(i\frac{mc^2}{\hbar} s \right) \:\Theta(-s))\\

\end{eqnarray}

with [itex]s^2 = c^2(t-t')^2-(\vec{x}-\vec{x}')^2[/itex], the source point [itex]x'[/itex], [itex]\Theta(s)[/itex] the Heaviside stepfunction, [itex]H_1^{(1)}(x)[/itex] the Hankel function of the first kind, and [itex]K_1(x)[/itex] the modified Bessel function of the second kind.

I changed to spherical polar coordinates and did the [itex]\phi[/itex] and [itex]\theta[/itex] integrals first. From there I've tried several approaches, but I always end up with a complex integral of a multivalued function:

\begin{equation}

\int \frac{f(z)}{\sqrt{z^2 \pm a^2}} dz

\end{equation}

From residues I get zero when [itex]f(z)[/itex] is an exponential function, but that can't be right?

Both my quantum field theory and many-particle books work in k-space and I haven't found the derivation here.

Would someone be kind enough to guide me throught the correct procedure for the different steps of the Fourier transform of the Feynman propagator?

Thanks for your time!