Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | |||
cauchy_principal_value [2014/02/22 18:57] nikolaj old revision restored (2014/02/22 18:53) |
cauchy_principal_value [2014/02/22 18:57] nikolaj |
||
---|---|---|---|
Line 2: | Line 2: | ||
==== 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: $\mathcal P\int_a^b: \mathbb R^2\times(\mathbb R\to\overline{\mathbb R})\to\overline{\mathbb R}$ | | ||
- | | @#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: $\mathcal P\int_a^b(f):=\mathrm{lim}_{\varepsilon\to 0}\left(\int_a^{p_1-\varepsilon}f(x)\,\mathrm dx+\int_{p_1+\varepsilon}^{p_2-\varepsilon}f(x)\,\mathrm dx+\cdots+\int_{p_m+\varepsilon}^b f(x)\,\mathrm dx\right)$ | | ||
==== Discussion ==== | ==== Discussion ==== |