Chapter 8.11, Problem 25P

Let

Show that $F^{\prime \prime }(x)=0$ for all $x\ne 0,$ and ${\displaystyle {\int }_{-\infty }^{\infty }{F}^{\prime \prime }}(x)dx=1,$ which leads you to think that $F^{\prime \prime }(x)$ might $=\delta (x).$ Show in two ways, as outlined in (a) and (b), that this is not true.

(a) Show that ${\displaystyle {\int }_{-\infty }^{\infty }\varphi }(x)F^{\prime \prime }(x)dx=\varphi (0)+2{\varphi }^{\prime }(0),$ where $\varphi$ is any test function. Then by (11.6) and $(11.14),$ what is $F^{\prime \prime }(x)?$

(b) Show that $F(x)=(x-2)u(x)$ where $u(x)$ is the unit step function in (11.17) Differentiate this equation twice and simplify using (11.17) and (11.18). Compare your result in (a).

(c) As in (a) and (b), find $G^{\prime \prime }(x)$ in terms of $\delta$ and ${\delta }^{\prime }$ if