summaryrefslogtreecommitdiff
path: root/source/know/concept/cauchy-principal-value
diff options
context:
space:
mode:
Diffstat (limited to 'source/know/concept/cauchy-principal-value')
-rw-r--r--source/know/concept/cauchy-principal-value/index.md18
1 files changed, 9 insertions, 9 deletions
diff --git a/source/know/concept/cauchy-principal-value/index.md b/source/know/concept/cauchy-principal-value/index.md
index a2582f2..f09611b 100644
--- a/source/know/concept/cauchy-principal-value/index.md
+++ b/source/know/concept/cauchy-principal-value/index.md
@@ -7,15 +7,15 @@ categories:
layout: "concept"
---
-The **Cauchy principal value** $\mathcal{P}$,
+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 $x = b$,
+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:
+To resolve this, we define the Cauchy principal value $$\mathcal{P}$$ as follows:
$$\begin{aligned}
\boxed{
@@ -24,8 +24,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-If $f(x)$ instead has a singularity at postive infinity $+\infty$,
-then we define $\mathcal{P}$ as follows:
+If $$f(x)$$ instead has a singularity at postive infinity $$+\infty$$,
+then we define $$\mathcal{P}$$ as follows:
$$\begin{aligned}
\boxed{
@@ -34,10 +34,10 @@ $$\begin{aligned}
}
\end{aligned}$$
-And analogously for $-\infty$.
-If $f(x)$ has singularities both at $+\infty$ and at $b$,
+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:
+such that $$\mathcal{P}$$ is given by:
$$\begin{aligned}
\mathcal{P} \int_{a}^\infty f(x) \:dx
@@ -49,5 +49,5 @@ And so on, until all problematic singularities have been dealt with.
In some situations, for example involving
the [Sokhotski-Plemelj theorem](/know/concept/sokhotski-plemelj-theorem/),
-the symbol $\mathcal{P}$ is written without an integral,
+the symbol $$\mathcal{P}$$ is written without an integral,
in which case the calculations are implicitly integrated.