Cauchy Principal Value#
Definition: Cauchy Principal Value
Let \(f: \mathbb{R} \to \mathbb{R}\) be a real function which is locally Riemann-integrable on \(\mathbb{R}\).
The Cauchy principal value of \(f\) is the following [limit](../Limits%20(Real%20Functions.md) of its Riemann integral (if it exists):
\[\lim_{M \to \infty} \int_{-M}^{+M} f(x)\,\mathrm{d}x\]
Notation
\[\text{CPV} \int_{-\infty}^{+\infty} f(x) \,\mathrm{d}x\]
Example: \(\text{CPV} \int_{-\infty}^{+\infty} x \,\mathrm{d}x\)
We want to determine the following Cauchy principal value:
\[\text{CPV} \int_{-\infty}^{+\infty} x \,\mathrm{d}x\]
We have:
\[\begin{aligned}\text{CPV} \int_{-\infty}^{+\infty} x \,\mathrm{d}x & = \lim_{M \to \infty} \int_{-M}^M x \,\mathrm{d}x \\ & = \lim_{M \to \infty} \left. \frac{1}{2}x^{2} \right\vert_{-M}^M \\ & = \lim_{M \to \infty} \left(\frac{1}{2}M^2 - \frac{1}{2}(-M)^2 \right) \\ & = \lim_{M \to \infty} 0 = 0\end{aligned}\]
Theorem: Improper Integral \(\implies\) Cauchy Principal Value
Let \(f: \mathbb{R} \to \mathbb{R}\) be a real function.
If \(f\) is improperly Riemann-integrable on \((-\infty, +\infty)\), then its Cauchy principal value exists and is equal to the integral of \(f\):
\[\int_{-\infty}^{+\infty} f(x) \,\mathrm{d}x = \text{CPV} \int_{-\infty}^{+\infty} f(x) \,\mathrm{d}x\]
Proof
TODO