# How to solve this improper integral - partial fractions

I'm trying to solve the undefined integral (when $$E=E'$$):

$$\int\frac{\left|\xi_{U}\right|^{2}}{\left(E'-U\right)\left(E-U\right)}dU=\int\frac{\vert\xi_{U}\vert^{2}}{(E-U)^{2}}dU$$

For that, i have to transform my integral in:

$$\int\frac{\vert\xi_{U}\vert^{2}}{(E-U)^{2}}dU=\lim_{E^{\prime}\rightarrow E}\frac{1}{E-E^{\prime}+2i0^{+}}\left(\int\frac{\vert\xi_{U}\vert^{2}}{E^{\prime}-U-i0^{+}}dU-\int\frac{\vert\xi_{U}\vert^{2}}{E-U+i0^{+}}dU\right)$$

To solve this, is used the relation:

$$\frac{1}{x\pm i0^{+}}={\cal P}\frac{1}{x}\mp i\pi\delta(x)$$

So i get:

$$\int\frac{\vert\xi_{U}\vert^{2}}{(E-U)^{2}}dU=\lim_{E^{\prime}\rightarrow E}\frac{1}{E-E^{\prime}+2i0^{+}}\left({\cal P}\int\frac{\vert\xi_{U}\vert^{2}}{E^{\prime}-U}dU+i\pi\vert\xi_{E'}\vert^{2}-\left({\cal P}\int\frac{\vert\xi_{U}\vert^{2}}{E-U}dU-i\pi\vert\xi_{E}\vert^{2}\right)\right)$$ $$=\lim_{E^{\prime}\rightarrow E}\frac{1}{E-E^{\prime}+2i0^{+}}\left({\cal P}\int\frac{\vert\xi_{U}\vert^{2}}{E^{\prime}-U}dU+i\pi\vert\xi_{E'}\vert^{2}-{\cal P}\int\frac{\vert\xi_{U}\vert^{2}}{E-U}dU+i\pi\vert\xi_{E}\vert^{2}\right)$$

Now, i try to do a number of simplifications to get to the desired result: $$\int\frac{\vert\xi_{U}\vert^{2}}{(E-U)^{2}}dU=\lim_{E^{\prime}\rightarrow E}\frac{1}{E-E^{\prime}+2i0^{+}}\left(F(E')+i\pi\vert\xi_{E'}\vert^{2}-F(E)+i\pi\vert\xi_{E}\vert^{2}\right)$$

Where i called $$F(E)={\cal P}\int\frac{\vert\xi_{U}\vert^{2}}{E-U}dU$$. So, now: $$\int\frac{\vert\xi_{U}\vert^{2}}{(E-U)^{2}}dU=\lim_{E^{\prime}\rightarrow E}\frac{1}{2}\frac{1}{\frac{E-E^{\prime}}{2}+i0^{+}}\left(F(E')+i\pi\vert\xi_{E'}\vert^{2}-F(E)+i\pi\vert\xi_{E}\vert^{2}\right)$$ $$=\lim_{E^{\prime}\rightarrow E}\left({\cal P}\frac{1}{E-E'}-\frac{1}{2}i\pi\delta(\frac{E-E'}{2})\right)\left(F(E')+i\pi\vert\xi_{E'}\vert^{2}-F(E)+i\pi\vert\xi_{E}\vert^{2}\right)$$ $$=\lim_{E^{\prime}\rightarrow E}\left({\cal P}\frac{1}{E-E'}-i\pi\delta(E-E')\right)\left(F(E')+i\pi\vert\xi_{E'}\vert^{2}-F(E)+i\pi\vert\xi_{E}\vert^{2}\right)$$

Now, i get stuck in here. I'm supposed to get: $$\int\frac{\vert\xi_{U}\vert^{2}}{(E-U)^{2}}dU=-\lim_{E^{\prime}\rightarrow E}\frac{F(E)-F(E^{\prime})}{E-E^{\prime}}+\pi^{2}\vert\xi_{E}\vert^{2}\delta(E-E^{\prime}).$$