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Cauchy principal value
Partial function
| $\mathcal P\int_a^b: \mathbb R^2\times(\mathbb R\to\overline{\mathbb R})\to\overline{\mathbb R}$ |
| $\mathcal P\int_a^b(f):=\mathrm{lim}_{\varepsilon\to 0}\left(\int_a^{p_1-\varepsilon}f(x)\,\mathrm dx+\int_{p_1+\varepsilon}^{p_2-\varepsilon}f(x)\,\mathrm dx+\cdots+\int_{p_m+\varepsilon}^b f(x)\,\mathrm dx\right)$ |
| $p$ … ordered sequence of the $m$ poles of $f$ |
Discussion
The Cauchy principal value is the value of an integral where the singularities are approached in a symmetric fashion.
Examples
$\mathcal P\int_{-3}^7\left(\lambda x.\frac{1}{x}\right)=\int_{-3}^{-\varepsilon}\frac{1}{x}\,\mathrm dx+\int_{\varepsilon}^{7}\frac{1}{x}\,\mathrm dx=\left(\log(+\varepsilon)-\log(+3)\right)+\left(\log(7)-\log(\varepsilon)\right)=\log(\frac{7}{3})$
Reference
Wikipedia: Cauchy principal value
Parents
Requirements
Related
