In doing research on Beltrami equations I am attempting to familiarize myself with the Beurling Transform, which is given as a principal value. Given $\phi \in C_0^\infty(\mathbb{C})$ we define the transform as follows: $$ (\mathcal{S}\phi)(z) = -\frac{1}{\pi}\lim_{\epsilon \rightarrow 0}\int_{|z-\tau|>\epsilon}\frac{\phi(\tau)}{(z-\tau)^2}d\tau. $$
Given such nice functions $\phi$, the author notes that this limit exists for every $z \in \mathbb{C}$. I am struggling, however, to prove why such a limit exists. This is reminiscent of proving existence of such a principal value for the Hilbert transform. However in that case your kernel $1/x$ is odd and so you can utilize the fact that your kernel has mean value zero over an annular region to then cleverly use the mean value theorem. As we see I appear unable to do that here. Is there a more general way to show this limit exists?