Question: Are there arguments or examples that shows I should not always use Cauchy Principal Value when Riemann's integral is not defined for mathematical applications?
I take the case which $f(x)$ has a singularity at $x_0 \in \left(0, \ 1\right)$. Suppose that $f(x)$ can be expanded by Laurent series around $x_0$ by using the odd terms:
$$f(x) = \sum_{j=0}^{k}\dfrac{b_{2j+1}}{(x-x_0)^{2j+1}} + \sum_{j=0}^{\infty} a_j \cdot (x-x_0)^{j} $$
My interest is this integral
$$I = \int_{0}^{1} f(x) \ dx$$
Description:
I read some questions that discuss the existence (or not) of this type of integral
$$\int_{-1}^{1} \dfrac{1}{x} \ dx\tag{1}\label{1}$$
Which Cauchy Principal Value is given by \eqref{2}
$$\int_{-1}^{1} \dfrac{1}{x} \ dx = \lim_{\varepsilon \to 0^{+}} \int_{-1}^{-\varepsilon} \dfrac{1}{x} \ dx + \int_{\varepsilon}^{1} \dfrac{1}{x} \ dx = 0 \tag{2}\label{2}$$
And by using the primitive
$$f(x) = \dfrac{1}{x} \Rightarrow F(x) = \ln |x|$$ $$\int_{-1}^{1} f(x) \ dx = F(1)-F(-1) = \ln |1| - \ln |-1| = 0 \tag{3}\label{3}$$
- Like shown here, the main reason of non-existence of \eqref{1} is because the the limit of each part doesn't exist
$$\int_{-1}^{1} \dfrac{1}{x} \ dx = \int_{-1}^{0} \dfrac{1}{x} \ dx + \int_{0}^{1} \dfrac{1}{x} \ dx$$
- Here, they shifted the singularity along the imaginary axis and did the integration along the real axis. As the shifted function is holomorphic along the path, the integral is calculated as the primitive just like in \eqref{3}.
$$\int_{-1}^{1} \dfrac{1}{x} \ dx = \lim_{\varepsilon \to 0} \int_{-1}^{1} \dfrac{1}{x+i\varepsilon} \ dx = 0$$
- Here, there's an argument about the existence of \eqref{1} by using the difference of potential (in physics), which relates to the total work between $-1$ and $1$. It's reliable to \eqref{2} which limit order doesn't matter much.
So far, all arguments I've read against using CPV was about the mathematic definition of the integral. The Riemann's integral is defined only if $f(x)$ is bounded, which is not the case. Are there others arguments to not use CPV when Riemann's don't exist for mathematical applications?
