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cauchy_principal_value [2014/02/22 18:56] nikolaj |
cauchy_principal_value [2014/03/21 11:11] (current) |
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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 $f$ | | ||
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| + | | @#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 ==== | ||
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| === Examples === | === Examples === | ||
| - | $\mathcal P\int_{-3}^7\left(\lambda x.\frac{1}{x}\right)=\int_{-3}^{-\varepsilon}\frac{1}{x}\,\mathrm dx+\int_{\varepsilon}^{7}\frac{1}{x}\,\mathrm dx=\left(\log(+\varepsilon)-\log(+3)\right)+\left(\log(7)-\log(\varepsilon)\right)=\log(\frac{7}{3})$ | + | $\mathcal P\int_{-3}^7\left(\lambda x.\frac{1}{x}\right)=\int_{-3}^{-\varepsilon}\frac{1}{x}\,\mathrm dx+\int_{\varepsilon}^{7}\frac{1}{x}\,\mathrm dx=\left(\log\left|-\varepsilon\right|-\log\left|-3\right|\right)+\left(\log\left|7\right|-\log\left|\varepsilon\right|\right)=\log\left|\frac{7}{3}\right|$ |
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Cauchy_principal_value|Cauchy principal value]] | Wikipedia: [[http://en.wikipedia.org/wiki/Cauchy_principal_value|Cauchy principal value]] | ||
