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I'm trying to understand why the wavefront set of the distribution $1/(x+i0)$ is given by $\left\{(x,\xi):\xi=0\ \textrm{or}\ x=0,\xi\geq0\right\}$, and why that of $1/(x-i0)$ is $\left\{(x,\xi):\xi=0\ \textrm{or}\ x=0,\xi\leq0\right\}$.

Also, I'm trying to figure out the wavefront set of the principal value of $1/x$.

Can anyone please help?

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