diff options
author | Prefetch | 2022-10-14 23:25:28 +0200 |
---|---|---|
committer | Prefetch | 2022-10-14 23:25:28 +0200 |
commit | 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae (patch) | |
tree | a0abb6b22f77c0e84ed38277d14662412ce14f39 /source/know/concept/cauchy-principal-value |
Initial commit after migration from Hugo
Diffstat (limited to 'source/know/concept/cauchy-principal-value')
-rw-r--r-- | source/know/concept/cauchy-principal-value/index.md | 52 |
1 files changed, 52 insertions, 0 deletions
diff --git a/source/know/concept/cauchy-principal-value/index.md b/source/know/concept/cauchy-principal-value/index.md new file mode 100644 index 0000000..fec0826 --- /dev/null +++ b/source/know/concept/cauchy-principal-value/index.md @@ -0,0 +1,52 @@ +--- +title: "Cauchy principal value" +date: 2021-11-01 +categories: +- Mathematics +layout: "concept" +--- + +The **Cauchy principal value** $\mathcal{P}$, +or just **principal value**, +is a method for integrating problematic functions, +i.e. functions with singularities, +whose integrals would otherwise diverge. + +Consider a function $f(x)$ with a singularity at some finite $x = b$, +which is hampering attempts at integrating it. +To resolve this, we define the Cauchy principal value $\mathcal{P}$ as follows: + +$$\begin{aligned} + \boxed{ + \mathcal{P} \int_a^c f(x) \dd{x} + = \lim_{\varepsilon \to 0^{+}} \!\bigg( \int_a^{b - \varepsilon} f(x) \dd{x} + \int_{b + \varepsilon}^c f(x) \dd{x} \bigg) + } +\end{aligned}$$ + +If $f(x)$ instead has a singularity at postive infinity $+\infty$, +then we define $\mathcal{P}$ as follows: + +$$\begin{aligned} + \boxed{ + \mathcal{P} \int_{a}^\infty f(x) \dd{x} + = \lim_{c \to \infty} \!\bigg( \int_{a}^c f(x) \dd{x} \bigg) + } +\end{aligned}$$ + +And analogously for $-\infty$. +If $f(x)$ has singularities both at $+\infty$ and at $b$, +then we simply combine the two previous cases, +such that $\mathcal{P}$ is given by: + +$$\begin{aligned} + \mathcal{P} \int_{a}^\infty f(x) \:dx + = \lim_{c \to \infty} \lim_{\varepsilon \to 0^{+}} + \!\bigg( \int_{a}^{b - \varepsilon} f(x) \:dx + \int_{b + \varepsilon}^{c} f(x) \:dx \bigg) +\end{aligned}$$ + +And so on, until all problematic singularities have been dealt with. + +In some situations, for example involving +the [Sokhotski-Plemelj theorem](/know/concept/sokhotski-plemelj-theorem/), +the symbol $\mathcal{P}$ is written without an integral, +in which case the calculations are implicitly integrated. |