Expressing the Principal Value of 1/x

Question

I'm a graduate student and taking a course in analysis and came upon this definition for the cauchy principal value for 1/x, can someone kindly explain how this equality comes about and perhaps provide a literature link to this equality? Thank You :) $$ \lim_{a \to +\infty} \int_{\mathbb{R} } \frac{(1-cos(ax))\varphi(x) }{x} dx = p.v.(\frac{1}{x}) $$

Answer 1

In the sense of distributions, $\cos(ax) \to 0$ as $a \to \pm\infty$ because the close "peaks" and "valleys" will cancel. Therefore, $\frac{1-\cos(ax)}{x} \to \frac{1-0}{x} = \frac{1}{x}$ at least away from $x=0.$

At $x=0$ the expression removes the infinities since $$ \frac{1-\cos(ax)}{x} = \frac{1-(1-\frac12(ax)^2+O(x^4)}{x} = \frac12 a^2 x + O(x^3). $$

Probably the construction of the expression was based on the above facts.