I am trying to compute the Green function of $$ -\Delta u+u $$ using Fourier transform: $(|\xi|^2+1)\hat{u}=1$. Is there any reference where I can find this calculation?
thank you.
I started with Fubini: $\int_{\mathbb{R}^n}\frac{e^{i\xi\cdot x}}{1+|\xi|^2}d\xi=$ $\int_{\mathbb{R}^{n-1}}d\xi_{n-1}e^{i(x_2\xi_2+..+\xi_nx_n)}\int_{\mathbb{R}}\frac{e^{i\xi_1\cdot x_1}}{1+\xi_1^2+\xi_2^2+..+\xi_n^2}d\xi_1$
Using contour integration of $\int_{\gamma}\frac{e^{izx_1}}{z^2+s^2}dz$
where $s^2=1+\xi_2^2+..+\xi_n^2$
on half circle (whose radii $R\to\infty$) in the upper half plane. The result is
$\int_{\mathbb{R}}\frac{e^{i\xi_1\cdot x_1}}{1+\xi_1^2+\xi_2^2+..+\xi_n^2}d\xi_1=2\pi i \cdot f(is)$ where
$f(z)=\frac{e^{izx_1}}{z+is}$.
To continue forward the next integral ($d\xi_2$) I need to evaluate
$\int_{\mathbb{R}}\frac{e^{i\xi_2 x_2}\cdot e^{-sx_1}}{-\sqrt{1+\xi_2^2+..+\xi_n^2}} d\xi_2$
It looks very complicated. maybe there is a better way?
thank you