Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods?

Question

Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods?

In all of the examples I have seen, poles are simple and this helps to avoid those singular points since integrals around them converge to a finite value (such as $i\pi$) when the radius of the circular path around them goes to zero. (In nonsimple poles I guess the integral does not converge around them, on the circle with infinitesimal path.)

For example, for the integral $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ Wolfram alpha (correctly)gives $\pi$ as the principal value, but says $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x^3}dx$$

or

$$\int_{-\infty}^{\infty}\frac{\sin(x)}{x^4}dx$$ do not converge.

Answer 1

The first one is an ordinary improper integral, and Maple says

$ \int_{-\infty}^\infty \sin(x)/x $

is $\pi$. The others are, indeed, poles.

This one has a pole of order $1$, and Maple says:

$ \int_{-\infty}^\infty \sin(x)/x^2$, based on CauchyPrincipalValue);

is $0$ as we expect. (And the non-principal-value integral is "undefined".)

This one has a pole of order $2$, and Maple says

$ \int_{-\infty}^\infty \sin(x)/x^3$,,CauchyPrincipalValue=true);

is $\infty$.

And for higher integer powers, Maple has alternately $0$ and $\infty$ for the Cauchy principal values.