I'm reading the book Mathematical Physics, and here's a problem I couldn't figure out.
Let $\varphi(x)$ be a test function, show that
\begin{equation}\frac{d}{dt}\{P \int_{-\infty}^{\infty} \frac{\phi(x)}{(x-t)}d x\}=P\int_{-\infty}^\infty \frac{\phi(x)-\phi(t)}{(x-t)^2}\ dx.\end{equation}
The problem is, the only thing I know about principle value is that $$P(\frac{1}{x}):=\lim_{\epsilon\rightarrow 0}\frac{x}{x^2+\epsilon^2}=\left\{ \begin{aligned} &0,\quad x=0,\\ &\frac{1}{x}, \quad x\neq 0. \end{aligned} \right. $$ So I'm not sure what the principal value here actually means.
Thanks in advance for whoever could give me some advice.
Cheers!