---
title: Cauchy Principal Value
tags:
  - math
  - real-analysis
date: 2024-01-12
---
# Notation


$$
\text{p.v.} \int_{-\infty}^{\infty} f(x)dx = \lim_{a\rightarrow+\infty} \int_{-a}^{a} f(x) dx = \lim_{a\rightarrow+\infty}[f(a) - f(-a)]
$$



![](math/real_analysis/attachments/6BC0B163CEFCF127E1D70326AB7D1648%201.png)


![](math/real_analysis/attachments/78DC2683DB0DF2EFEB6215DAB8C18C25.png)

the Cauchy principal value is the method for assigning values to *certain improper integrals* which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain.

# Reference

* [_Real Analysis 64 | Cauchy Principal Value_. _www.youtube.com_, https://www.youtube.com/watch?v=0SP2b0nFpwI. Accessed 3 Jan. 2024.](https://www.youtube.com/watch?v=0SP2b0nFpwI)
* [“Cauchy Principal Value.” _Wikipedia_, 31 Dec. 2023. _Wikipedia_, https://en.wikipedia.org/w/index.php?title=Cauchy_principal_value&oldid=1192842366.](https://en.wikipedia.org/wiki/Cauchy_principal_value)