I'm trying to verify that $u(t,x)=H(t-|x|)(t^2-|x|^2)^{-1/2}$ is the fundamental solution of the 2-dimensional wave equation; that is, $\Box u = u_{tt}-\Delta u = \delta_{0}$. I know there are tricks that make this calculation easier (involving realizing distributions as limits) but I'd really like to figure out how to do this with straightforward integration techniques.
I need to show that $\langle u(t,x),\Box\varphi(t,x)\rangle = \langle\delta_0,\varphi(t,x)\rangle = \varphi(0,0,0),$ where $\varphi\in C_{0}^{\infty}(\mathbb{R}^{1+2})$. So far I have:
$\langle u(t,x),\Box\varphi(t,x)\rangle =\int_{-\infty}^{\infty}\int_{\mathbb{R}^2}H(t-|x|)(t^2-|x|^2)^{-1/2}\Box\varphi(t,x)dAdt$
$=\int_{-\infty}^{\infty}\int_{0}^{2\pi}\int_{0}^{\infty}H(t-r)(t^2-r^2)^{-1/2}r\Box_{\text{pol}}\varphi(t,r,\theta)drd\theta dt$ $=\int_{0}^{\infty}\int_{0}^{2\pi}\int_{0}^{t}\partial_{r}\left((t^2-r^2)^{\frac{1}{2}}\right)\left( \varphi_{tt} - \Delta_{\text{pol}}\varphi\right)drd\theta dt$ $=\int_{0}^{\infty}\int_{0}^{2\pi}\int_{0}^{t}\partial_{r}\left((t^2-r^2)^{\frac{1}{2}}\right)\left( \varphi_{tt}-\frac{1}{r}\partial_{r}\left(r\varphi_r\right)-\frac{1}{r^2}\varphi_{\theta\theta} \right)drd\theta dt.$
Can anyone point me in the right direction? I'm sure integration by parts will be important here, but I'm not sure how to proceed. Any help?