I did the computation of the propagator in two dimensions at (19.26) in Peskin & Shroeder as follows.
First I performed a Wick rotation.
\begin{alignat}{2} \int\frac{d^2 k}{(2\pi)^2}e^{-ik\cdot (y-z)}\frac{ik^{\mu}\gamma_{\mu}}{k^2} &=&& -\partial^{\mu}\gamma_{\mu} \left( i\int \frac{d^2 k_E}{(2\pi)^2}e^{ik_E\cdot (y_E-z_E)}\frac{1}{-k_E^2} \right) \\ &=&& \frac{i}{4\pi^2} \partial^{\mu}\gamma_{\mu} \int_0^{\infty}dk_E k_E \frac{1}{k_E^2}\int_0^{2\pi}d\theta e^{ik_E|y_E-z_E|\cos \theta} \\ &=&& \frac{i}{4\pi^2} \partial^{\mu}\gamma_{\mu} \int_0^{\infty}dk_E k_E \frac{1}{k_E^2} 2\pi J_0(k_E|y_E-z_E|) \\ \end{alignat}
where $J_0(s)$ is a bessel function and I made use of Hansen-Bessel Formula.
Setting $s\equiv k_E|y_E-z_E|$
\begin{alignat}{2} &=&& \frac{i}{2\pi} \partial^{\mu}\gamma_{\mu} \int_0^{\infty} ds\frac{1}{s} J_0(s) \\ &=&& 0 \end{alignat} But in the book, $$ \int\frac{d^2 k}{(2\pi)^2}e^{-ik\cdot (y-z)}\frac{ik^{\mu}\gamma_{\mu}}{k^2}= -\partial^{\mu}\gamma_{\mu} \left( \frac{i}{4\pi}\log (y-z)^2 \right) \tag{19.26}$$ Where did I make a mistake?