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During a graduate course on Electrodynamics I began reviewing the Kramer-Kronig relations, which are defined using the Cauchy principal value. However, I have some trouble understanding how to evaluate the Cauchy principal value, since it's a topic of complex analysis I never saw before and so far the articles and books I've read just offer the formal definition without examples.

First of all, I understand that if $h(z)$ is singular in $z=0$, then the principal value can be evaluated as [1],

$$Pint_-infty^+infty h(z)dz=lim_arightarrow 0^+left[ int_-infty^-a h(z)dz+int_a^+infty h(z)dzright]$$

However, this only applies if $h(z)$ has one pole in $z=0$. If, for example, I have the function,

$$int_-infty^+infty fracomegaz(z-z_0)dz$$

How is the principal value defined in this case?

I have also found in other documents that the definition of the principal value is a bit different, as for example, in [2] they define the principal value as,

$$Pint_-infty^+inftyh(z)dz=lim_Rrightarrowinftyint_-R^+Rh(z)dz$$

which can be evaluated as usual using the residue theorem, so I'm confused why there're are two definitions, one of which is actually familiar.

[1] https://math.tutorvista.com/calculus/cauchy-principal-value.html

[2] http://stat.math.uregina.ca/~kozdron/Teaching/Regina/312Fall13/Handouts/lecture34_dec_2.pdf

Both are referred to as the "Principal Value", and give the practical result we would expect from contour integration. Both forms can be seen as a specialization of
$$P.V.int_-infty^infty f = lim_varepsilon rightarrow 0^+ left[int_b-frac1varepsilon^b-varepsilon f(x),mathrmdx+int_b+varepsilon^b+frac1varepsilonf(x),mathrmdx right]$$

whenever $f$ has a singularity at $b,$ but even then we need a more general definition when $f$ has more singularities. A better definition is to simply define the Principal Value by
$$P.V.int_C f(z) mathrmdz = lim_varepsilon to 0^+ int_C setminus mathcalN_epsilon (b) f(z) mathrmdz$$ where $C$ is a contour in the complex plane and $mathcalN_epsilon (b)$ is ball of radius $epsilon$ about a singularity $b$ of $f.$ Since complex singularities are isolated, we can obviously break up our integral and handle each singularity separately, so this is fully general.