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cauchy_principal_value [2014/02/22 18:57] nikolaj old revision restored (2014/02/22 18:53) |
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| ===== Cauchy principal value ===== | ===== Cauchy principal value ===== | ||
| ==== Partial function ==== | ==== Partial function ==== | ||
| - | | @#FFBB00: $\mathcal P\int_a^b: \mathbb R^2\times(\mathbb R\to\overline{\mathbb R})\to\overline{\mathbb R}$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\mathcal P\int_a^b: \mathbb R^2\times(\mathbb R\to\overline{\mathbb R})\to\overline{\mathbb R}$ | |
| - | | @#FFBB00: $\mathcal P\int_a^b(f):=\mathrm{lim}_{\varepsilon\to 0}\left(\int_a^{p_1-\varepsilon}f(x)\,\mathrm dx+\int_{p_1+\varepsilon}^{p_2-\varepsilon}f(x)\,\mathrm dx+\cdots+\int_{p_m+\varepsilon}^b f(x)\,\mathrm dx\right)$ | | + | |
| | @#BBDDEE: $p$ ... ordered sequence of the $m$ poles of $f$ | | | @#BBDDEE: $p$ ... ordered sequence of the $m$ poles of $f$ | | ||
| + | |||
| + | | @#FFBB00: definiendum | @#FFBB00: $\mathcal P\int_a^b(f):=\mathrm{lim}_{\varepsilon\to 0}\left(\int_a^{p_1-\varepsilon}f(x)\,\mathrm dx+\int_{p_1+\varepsilon}^{p_2-\varepsilon}f(x)\,\mathrm dx+\cdots+\int_{p_m+\varepsilon}^b f(x)\,\mathrm dx\right)$ | | ||
| ==== Discussion ==== | ==== Discussion ==== | ||
