Cauchy principal value

Partial function

definiendum

$\mathcal P\int_a^b: \mathbb R^2\times(\mathbb R\to\overline{\mathbb R})\to\overline{\mathbb R}$

$p$ … ordered sequence of the

$m$ poles of

$f$

definiendum

$\mathcal P\int_a^b(f):=\mathrm{lim}_{\varepsilon\to 0}\left(\int_a^{p_1-\varepsilon}f(x)\,\mathrm dx+\int_{p_1+\varepsilon}^{p_2-\varepsilon}f(x)\,\mathrm dx+\cdots+\int_{p_m+\varepsilon}^b f(x)\,\mathrm dx\right)$

Discussion

The Cauchy principal value is the value of an integral where the singularities are approached in a symmetric fashion.

Examples

$\mathcal P\int_{-3}^7\left(\lambda x.\frac{1}{x}\right)=\int_{-3}^{-\varepsilon}\frac{1}{x}\,\mathrm dx+\int_{\varepsilon}^{7}\frac{1}{x}\,\mathrm dx=\left(\log\left|-\varepsilon\right|-\log\left|-3\right|\right)+\left(\log\left|7\right|-\log\left|\varepsilon\right|\right)=\log\left|\frac{7}{3}\right|$

Reference

Wikipedia: Cauchy principal value

Parents

Requirements

Lebesgue measure

Pole of a complex function