(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that, in a distributional sense,

x* Pv(1/x) = 1

2. Relevant equations

The function 1/x cannot be integrated locally in the origin. Nevertheless,

int(1/x, x=-1..1) =0, and thus convergent.

Therefore, one defines the (non-regular) distribution Pv(1/x), as follows:

<Pv(1/x), phi(x)> = Pv*int(phi(x)/x, x=-infinity..infinity)

which is defined to be equal to:

limit( eps->0) ( int( phi(x)/x,x=-infinity..-eps) + int( phi(x)/x, x=eps..infinity) )

3. The attempt at a solution

My problem is that I don't really understand how I can write x*Pv(1/x) as a distribution. My attempt at a solution is as follows::

<x*Pv(1/x), phi(x)> = limit(eps->0) (int( phi(x)/x*x,x=-infinity..-eps) = int( phi(x)/x*x, x=eps..infinity)

If this step is allowed, then of course, this integration is equal to :

int(phi(x),x=-infinity..infinity) = 1 , because this is a distribution. But surely, it can't be as simple as that!

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# Principle Value of 1/x

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