Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see <http://adsabs.harvard.edu/abs/1947pqm..book.....D> for the exact reference (but no text). What is the best way of formalizing this to a mathematician's satisfaction?



Does the derivative of log have a Dirac delta term?
In integral form, this amounts to the  [Sokhotski-Plemelj](http://en.wikipedia.org/wiki/Sokhotski-Plemelj_theorem) theorem:


$\lim\_{\epsilon\rightarrow 0^{+}}\int\_{-\infty}^{\infty}dx f(x) \frac{d}{dx}\log (x+i\epsilon)=-i\pi f(0)+{\cal P}\int\_{-\infty}^{\infty}dx f(x)\frac{1}{x}$.


The symbol ${\cal P}$ indicates that the Cauchy principal value of the integral is to be taken. Formalization then amounts to stating the conditions on $f$ so that the principal value integral exists.


The logarithm for $x<0$ is defined as $\log x= \log(-x) + i\pi$. You can avoid the delta function by including the absolute value signs in the logarithm:


$\int\_{-\infty}^{\infty}dx f(x) \frac{d}{dx}\log|x|={\cal P}\int\_{-\infty}^{\infty}dx f(x)\frac{1}{x}$.


