From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/legendre-polynomials/index.md | 38 +++++++++++------------
 1 file changed, 19 insertions(+), 19 deletions(-)

(limited to 'source/know/concept/legendre-polynomials')

diff --git a/source/know/concept/legendre-polynomials/index.md b/source/know/concept/legendre-polynomials/index.md
index 74543a3..f223cd3 100644
--- a/source/know/concept/legendre-polynomials/index.md
+++ b/source/know/concept/legendre-polynomials/index.md
@@ -8,10 +8,10 @@ layout: "concept"
 ---
 
 The **Legendre polynomials** are a set of functions that sometimes arise in physics.
-They are the eigenfunctions $u(x)$ of **Legendre's differential equation**,
+They are the eigenfunctions $$u(x)$$ of **Legendre's differential equation**,
 which is a ([Sturm-Liouville](/know/concept/sturm-liouville-theory/))
-eigenvalue problem for $\ell (\ell + 1)$,
-where $\ell$ turns out to be a non-negative integer:
+eigenvalue problem for $$\ell (\ell + 1)$$,
+where $$\ell$$ turns out to be a non-negative integer:
 
 $$\begin{aligned}
     \boxed{
@@ -19,7 +19,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The $\ell$th-degree Legendre polynomial $P_\ell(x)$
+The $$\ell$$th-degree Legendre polynomial $$P_\ell(x)$$
 is given in the form of a *Rodrigues' formula* by:
 
 $$\begin{aligned}
@@ -27,7 +27,7 @@ $$\begin{aligned}
     &= \frac{1}{2^\ell \ell!} \dvn{\ell}{}{x}(x^2 - 1)^\ell
 \end{aligned}$$
 
-The first handful of Legendre polynomials $P_\ell(x)$ are therefore as follows:
+The first handful of Legendre polynomials $$P_\ell(x)$$ are therefore as follows:
 
 $$\begin{gathered}
     P_0(x) = 1
@@ -41,7 +41,7 @@ $$\begin{gathered}
     P_4(x) = \frac{1}{8} (35 x^4 - 30 x^2 + 3)
 \end{gathered}$$
 
-And then more $P_\ell$ can be computed quickly
+And then more $$P_\ell$$ can be computed quickly
 using **Bonnet's recursion formula**:
 
 $$\begin{aligned}
@@ -50,7 +50,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The derivative of a given $P_\ell$ can be calculated recursively
+The derivative of a given $$P_\ell$$ can be calculated recursively
 using the following relation:
 
 $$\begin{aligned}
@@ -61,7 +61,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Noteworthy is that the Legendre polynomials
-are mutually orthogonal for $x \in [-1, 1]$:
+are mutually orthogonal for $$x \in [-1, 1]$$:
 
 $$\begin{aligned}
     \boxed{
@@ -74,7 +74,7 @@ $$\begin{aligned}
 As was to be expected from Sturm-Liouville theory.
 Likewise, they form a complete basis in the
 [Hilbert space](/know/concept/hilbert-space/)
-of piecewise continuous functions $f(x)$ on $x \in [-1, 1]$,
+of piecewise continuous functions $$f(x)$$ on $$x \in [-1, 1]$$,
 meaning:
 
 $$\begin{aligned}
@@ -85,11 +85,11 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Each Legendre polynomial $P_\ell$ comes with
-a set of **associated Legendre polynomials** $P_\ell^m(x)$
-of order $m$ and degree $\ell$.
+Each Legendre polynomial $$P_\ell$$ comes with
+a set of **associated Legendre polynomials** $$P_\ell^m(x)$$
+of order $$m$$ and degree $$\ell$$.
 These are the non-singular solutions of the **general Legendre equation**,
-where $m$ and $\ell$ are integers satisfying $-\ell \le m \le \ell$:
+where $$m$$ and $$\ell$$ are integers satisfying $$-\ell \le m \le \ell$$:
 
 $$\begin{aligned}
     \boxed{
@@ -97,17 +97,17 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The $\ell$th-degree $m$th-order associated Legendre polynomial $P_\ell^m$
-is as follows for $m \ge 0$:
+The $$\ell$$th-degree $$m$$th-order associated Legendre polynomial $$P_\ell^m$$
+is as follows for $$m \ge 0$$:
 
 $$\begin{aligned}
     P_\ell^m(x)
     = (-1)^m (1 - x^2)^{m/2} \dvn{m}{}{x}P_\ell(x)
 \end{aligned}$$
 
-Here, the $(-1)^m$ in front is called the **Condon-Shortley phase**,
+Here, the $$(-1)^m$$ in front is called the **Condon-Shortley phase**,
 and is omitted by some authors.
-For negative orders $m$,
+For negative orders $$m$$,
 an additional constant factor is necessary:
 
 $$\begin{aligned}
@@ -115,6 +115,6 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Beware, the name is misleading:
-if $m$ is odd, then $P_\ell^m$ is actually not a polynomial.
-Moreover, not all $P_\ell^m$ are mutually orthogonal
+if $$m$$ is odd, then $$P_\ell^m$$ is actually not a polynomial.
+Moreover, not all $$P_\ell^m$$ are mutually orthogonal
 (but some are).
-- 
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