# Convolution of 1D Greens functions

So I need some help calculating the convolution of two 1D Greens functions, where the specific formula reads

$$\int dk G_k(w+i\eta)G_{k+q}(w+i\eta) = \int dk\left[\Im\left[ \frac{1}{(\omega+i\eta+k)(\omega+i\eta+k+q)}\right] \right]\;,$$

wher $\eta$ is an infinitesimal parameter,i.e., I am using the retarded Greens functions. The integration is over the real numbers and $\Im$ is the imaginary part.

From numerics and physical expectations I am expecting something proportional to $-\delta (q)$, but I can't for the life of me figure out how to correctly calculate the $q=0$ case. For the $q\neq 0$-case the residue theorem seems to yield the correct result (zero), but for $q=0$ it does not seem to work.

Has anybody dealt with something similar? I would appreciate any help.

• Welcome to Physics Stack Exchange. Please note that clear questions are essential for getting good answers. In this post we see an integral, but we don't know what it is. Is it the Green's function? Please edit the post to make it as clear as you would want a question posed to you to be. Also note that this is not a problem solving service. We require that questions ask a specific conceptual question. Posts asking for general help solving specific problems are closed under our "homework" policy, which you should read about in the help center. – DanielSank Dec 1 '15 at 22:35
• I am sorry for the incomplete post. I added the additional information. – PythonSparse Dec 1 '15 at 22:40
• Might Mathematics be better suited for this math question? – Kyle Kanos Dec 1 '15 at 22:45
• Maybe you're right. My hope was that some high energy physicist might have worked with something similar. – PythonSparse Dec 1 '15 at 22:46
• Whether to put questions like this here or on the math site is a really common dilemma. Since you can really just post this as a pure math problem I'd do that (removing all references to the physics) and post on math. Folks there are pretty dang good at integrals :D – DanielSank Dec 1 '15 at 22:59