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/laguerre-polynomials/index.md | 34 +++++++++++------------
 1 file changed, 17 insertions(+), 17 deletions(-)

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

diff --git a/source/know/concept/laguerre-polynomials/index.md b/source/know/concept/laguerre-polynomials/index.md
index fd3deb6..ba68343 100644
--- a/source/know/concept/laguerre-polynomials/index.md
+++ b/source/know/concept/laguerre-polynomials/index.md
@@ -8,8 +8,8 @@ layout: "concept"
 ---
 
 The **Laguerre polynomials** are a set of useful functions that arise in physics.
-They are the non-singular eigenfunctions $u(x)$ of **Laguerre's equation**,
-with the corresponding eigenvalues $n$ being non-negative integers:
+They are the non-singular eigenfunctions $$u(x)$$ of **Laguerre's equation**,
+with the corresponding eigenvalues $$n$$ being non-negative integers:
 
 $$\begin{aligned}
     \boxed{
@@ -17,7 +17,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The $n$th-order Laguerre polynomial $L_n(x)$
+The $$n$$th-order Laguerre polynomial $$L_n(x)$$
 is given in the form of a *Rodrigues' formula* by:
 
 $$\begin{aligned}
@@ -27,7 +27,7 @@ $$\begin{aligned}
     &= \frac{1}{n!} \Big( \dv{}{x}- 1 \Big)^n x^n
 \end{aligned}$$
 
-The first couple of Laguerre polynomials $L_n(x)$ are therefore as follows:
+The first couple of Laguerre polynomials $$L_n(x)$$ are therefore as follows:
 
 $$\begin{gathered}
     L_0(x) = 1
@@ -39,8 +39,8 @@ $$\begin{gathered}
 
 Based on Laguerre's equation,
 **Laguerre's generalized equation** is as follows,
-with an arbitrary real (but usually integer) parameter $\alpha$,
-and $n$ still a non-negative integer:
+with an arbitrary real (but usually integer) parameter $$\alpha$$,
+and $$n$$ still a non-negative integer:
 
 $$\begin{aligned}
     \boxed{
@@ -48,10 +48,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Its solutions, denoted by $L_n^\alpha(x)$,
+Its solutions, denoted by $$L_n^\alpha(x)$$,
 are the **generalized** or **associated Laguerre polynomials**,
 which also have a Rodrigues' formula.
-Note that if $\alpha = 0$ then $L_n^\alpha = L_n$:
+Note that if $$\alpha = 0$$ then $$L_n^\alpha = L_n$$:
 
 $$\begin{aligned}
     L_n^\alpha(x)
@@ -60,7 +60,7 @@ $$\begin{aligned}
     &= \frac{x^{-\alpha}}{n!} \Big( \dv{}{x}- 1 \Big)^n x^{n + \alpha}
 \end{aligned}$$
 
-The first couple of associated Laguerre polynomials $L_n^\alpha(x)$ are therefore as follows:
+The first couple of associated Laguerre polynomials $$L_n^\alpha(x)$$ are therefore as follows:
 
 $$\begin{aligned}
     L_0^\alpha(x) = 1
@@ -70,7 +70,7 @@ $$\begin{aligned}
     L_2^\alpha(x) = \frac{1}{2} (x^2 - 2 \alpha x - 4 x + \alpha^2 + 3 \alpha + 2)
 \end{aligned}$$
 
-And then more $L_n^\alpha$ can be computed quickly
+And then more $$L_n^\alpha$$ can be computed quickly
 using the following recurrence relation:
 
 $$\begin{aligned}
@@ -91,8 +91,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Noteworthy is that these polynomials (both normal and associated)
-are all mutually orthogonal for $x \in [0, \infty[$,
-with respect to the weight function $w(x) \equiv x^\alpha \exp(-x)$:
+are all mutually orthogonal for $$x \in [0, \infty[$$,
+with respect to the weight function $$w(x) \equiv x^\alpha \exp(-x)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -102,11 +102,11 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $\delta_{nm}$ is the Kronecker delta.
+Where $$\delta_{nm}$$ is the Kronecker delta.
 Moreover, they form a basis in
 the [Hilbert space](/know/concept/hilbert-space/)
-of all functions $f(x)$ for which $\Inprod{f}{w f}$ is finite.
-Any such $f$ can thus be expanded as follows:
+of all functions $$f(x)$$ for which $$\Inprod{f}{w f}$$ is finite.
+Any such $$f$$ can thus be expanded as follows:
 
 $$\begin{aligned}
     \boxed{
@@ -116,8 +116,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Finally, the $L_n^\alpha(x)$ are related to
-the [Hermite polynomials](/know/concept/hermite-polynomials/) $H_n(x)$ like so:
+Finally, the $$L_n^\alpha(x)$$ are related to
+the [Hermite polynomials](/know/concept/hermite-polynomials/) $$H_n(x)$$ like so:
 
 $$\begin{aligned}
     H_{2n(x)} &= (-1)^n 2^{2n} n! \: L_n^{-1/2}(x^2)
-- 
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