What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,


$ PV \int\_a^b f(t) dt = ? $,


where the integral is convergent in the upper limit, but divergent in the lower.
Thank you for any ideas and/or references.



One-sided Cauchy principal value
Let me give an example: you want to define a distribution on $\mathbb R$ which coincides with $1/t$ on $(0,+\infty)$ and vanishes on $(-\infty,0)$. Let us take
$$
T=\frac{d}{dt}(H(t)\ln t),\quad H=1\_{\mathbb R\_+},
$$
with a distributional derivative. You can easily generalize to a 1D situation with a finite number of poles of finite type. For instance, to define a distribution on $\mathbb R$ which coincides with $1/t^2$ on $(0,+\infty)$ and vanishes on $(-\infty,0)$, you take
$$
-\frac{dT}{dt}.
$$ 


