Derivative of absolute distribuion

Question

$(T',\phi) = -(T,\phi')$ is the definition of derivative of distribution function $T$ How to use this to evaluate:

1. $e^{|x|}$

2. $\sin|x|$

3. P($\frac{1}{x}$)

In 3, it is the cauchy principal value. Can I just use any test function? or I have to use specific ones?

Answer 1

If you want to go by definition, then you should take arbitrary test functions.

For the first distribution, you take a test function $\phi$, suppose that its support lies in $[-R,R]$, then, by definition,

$$\int_{\mathbb R} e^{|x|}\phi'(x)dx = \int_{-R}^0 e^{|x|}\phi'(x)dx+\int_{0}^R e^{|x|}\phi'(x)dx = \int_{-R}^0 e^{-x}\phi'(x)dx+\int_{0}^R e^{x}\phi'(x)dx $$ You integrate by parts, because all functions are regular:

$$\int_{-R}^0 e^{-x}\phi'(x)dx = \phi(0) + \int_{-R}^0 e^{-x}\phi(x)dx$$ $$\int_0^R e^{x}\phi'(x)dx = -\phi(0) - \int_0^R e^{x}\phi(x)dx,$$hence $$\int_{\mathbb R} e^{|x|}\phi'(x)dx = -\int_{-R}^R sgn(x)e^{|x|}\phi(x)dx= -\int_{\mathbb R} sgn(x)e^{|x|}\phi(x)dx,$$thus, by definition, $$(\exp(|x|))' = sgn(x)\exp(|x|).$$

The same technique applies for other distributions that you mentioned.