Need to solve this integral: $$I=\int_{-1}^{1}dx(\lim_{\varepsilon\to 0^+}\frac{\varepsilon}{\varepsilon^2+x^2}f(x)+\pi\vartheta(x)\frac{df(x)}{dx}(x)) $$ I think I should recognize the limit as a distribution, but I can't find which one with a precise argument, I think it is $ \delta$ Dirac function. So if it is correct this is my solution: $$I=\int_{-1}^{1}\delta(x)f(x)dx+\pi\int_{-1}^{1}\vartheta(x)\frac{df(x)}{dx}(x)dx=$$ $$f(0)+\pi(f(1)-f(0))$$
Is this correct? If the identification of the limit with delta function is correct, can you please tell me why?
Thanks a lot
Almost. The limit is $\pi \delta(x)$ since $$ \int_{-\infty}^{\infty} \frac{\epsilon}{\epsilon^2+x^2} \phi(x) \, dx = \{ x = \epsilon y \} = \int_{-\infty}^{\infty} \frac{\epsilon}{\epsilon^2+\epsilon^2 y^2} \phi(\epsilon y) \, \epsilon \, dy = \int_{-\infty}^{\infty} \frac{1}{1+y^2} \phi(\epsilon y) \, dy \\ \to \int_{-\infty}^{\infty} \frac{1}{1+y^2} \phi(0) \, dy = \left( \int_{-\infty}^{\infty} \frac{1}{1+y^2} \, dy \right) \phi(0) = \pi \phi(0) = \int_{-\infty}^{\infty} \pi \delta(x) \, \phi(x) \, dx $$ for all $\phi \in C_c^\infty.$