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],
$$P\int_{-\infty}^{+\infty} h(z)dz=\lim_{a\rightarrow 0^+}\left[ \int_{-\infty}^{-a} h(z)dz+\int_a^{+\infty} h(z)dz\right]$$
However, this only applies if $h(z)$ has one pole in $z=0$. If, for example, I have the function,
$$\int_{-\infty}^{+\infty} \frac{\omega}{z(z-z_0)}dz$$
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,
$$P\int_{-\infty}^{+\infty}h(z)dz=\lim_{R\rightarrow\infty}\int_{-R}^{+R}h(z)dz$$
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
Best Answer
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 $$P.V.\int_{-\infty}^\infty f = \lim_{\varepsilon \rightarrow 0^+} \left[\int_{b-\frac{1}{\varepsilon}}^{b-\varepsilon} f(x)\,\mathrm{d}x+\int_{b+\varepsilon}^{b+\frac{1}{\varepsilon}}f(x)\,\mathrm{d}x \right]$$
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 $$P.V.\int_{C} f(z) \ \mathrm{d}z = \lim_{\varepsilon \to 0^+} \int_{C \setminus \mathcal{N}_\epsilon (b)} f(z)\ \mathrm{d}z$$ where $C$ is a contour in the complex plane and $\mathcal{N}_\epsilon (b)$ 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.