Exercise 21, § 3

a) Let $f\in \mathscr{S}(\mathbf{R})$. The function $g$ on $\mathbf{R}$ such that $g(0) = 2f'(0)$ and $g(x) =$ $(f(x)-f(-x))/x$ for every $x\not = 0$ is integrable on $\mathbf{R}$ with respect to Lebesgue measure. The mapping which associates to $f$ the integral of $g$ over $\mathbf{R}$ is a tempered distribution, denoted vp, and called the Cauchy principal value of $1/x$.

b) For every $f\in \mathscr{S}(\mathbf{R})$, one has

$\langle$vp$, f\rangle =$ lim$_{\varepsilon\rightarrow 0}\int_{\mathbf{R}-[-\varepsilon ,\varepsilon]}\frac{f(x)}{x}dx$.

c) The restriction of vp to $\mathbf{R}-\{0\}$ is the distribution associated with the locally integrable function $x\mapsto 1/x$ on $\mathbf{R}-\{0\}$.

d) The distribution vp coincides with the derivative of the distribution associated with the locally integrable function on $\mathbf{R}$ taking value 0 at 0 and defined for $x\not = 0$ by $x\mapsto$ log($|x|$).

e) Let $\varepsilon >0$. Denote by $p_{\varepsilon}$ the function $x\mapsto 1/(x+i\varepsilon )$ on $\mathbf{R}$. One has

lim$_{\varepsilon\rightarrow 0}p_{\varepsilon}=$ vp $-i\pi \varepsilon_0$

in the space $\mathscr{S}'(\mathbf{R})$ endowed with the weak topology, where $\varepsilon_0$ is the tempered distribution associated with the point mass at 0.