I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression
$$G^+(\mathbf{0})=\frac{2m}{(2\pi)^2}\int d\mathbf{p}\ \frac{\Theta(\Lambda^2-\mathbf{p}^2)}{\mathbf{p}^2-\mathbf{k}^2-i\epsilon}$$
Where the heaviside function is introduced as a cut-off parameter. I started by evaluating the contour where the poles are located at $\mathbf{p}=\pm(\mathbf{k}-i\epsilon)$. I factored the denominator, but I am stuck on how to evaluate the residue given the Heaviside function in the argument:
$$Res\{G^+(\mathbf{0})\}=\lim_{p\to p^+}({\mathbf{p}-\mathbf{k}-i\epsilon})\frac{\Theta(\Lambda^2-\mathbf{p}^2)}{(\mathbf{p}-\mathbf{k}-i\epsilon)(\mathbf{p}+\mathbf{k}-i\epsilon)}=\frac{\Theta(\Lambda^2-\mathbf{p}^2)}{2\mathbf{k}-2i\epsilon}$$
Where the $2i\epsilon$ term will go to zero after taking the limit $\epsilon\rightarrow0$.
The answer involves a $log(\frac{\Lambda^2}{-\mathbf{k}^2})$, which confuses me more as to how I should evaluate this integral. Any help is appreciated!