Question: Cauchy principal value.... wrong answer? or missing something?


> restart:
> int(exp(I*x)/x,x=-infinity..infinity,CauchyPrincipalValue)
                                                                                                   Pi I

> int(exp(I*3*x)/x,x=-infinity..infinity,CauchyPrincipalValue)
                                                                                                   Pi I

> int(exp(I*k*x)/x,x=-infinity..infinity,CauchyPrincipalValue) assuming k::positive;
                                                                                                  2 I Pi

I would expect the last line to also give me Pi I

Mathematica gives a Pi I via:

FullSimplify[Integrate[Exp[I k x]/x, {x, -\[Infinity], \[Infinity]}, PrincipalValue->True, Assumptions->{k>0}],  Assumptions->{k>0}]

Any idea what this could be?


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