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-rw-r--r--source/know/concept/hermite-polynomials/index.md20
1 files changed, 10 insertions, 10 deletions
diff --git a/source/know/concept/hermite-polynomials/index.md b/source/know/concept/hermite-polynomials/index.md
index ce34030..2eb8e06 100644
--- a/source/know/concept/hermite-polynomials/index.md
+++ b/source/know/concept/hermite-polynomials/index.md
@@ -15,10 +15,10 @@ although slightly different definitions are used in those fields.
## Physicists' definition
-The **Hermite equation** is an eigenvalue problem for $n$,
-and the Hermite polynomials $H_n(x)$ are its eigenfunctions $u(x)$,
-subject to the boundary condition that $u$ grows at most polynomially,
-in which case the eigenvalues $n$ are non-negative integers:
+The **Hermite equation** is an eigenvalue problem for $$n$$,
+and the Hermite polynomials $$H_n(x)$$ are its eigenfunctions $$u(x)$$,
+subject to the boundary condition that $$u$$ grows at most polynomially,
+in which case the eigenvalues $$n$$ are non-negative integers:
$$\begin{aligned}
\boxed{
@@ -26,7 +26,7 @@ $$\begin{aligned}
}
\end{aligned}$$
-The $n$th-order Hermite polynomial $H_n(x)$
+The $$n$$th-order Hermite polynomial $$H_n(x)$$
is therefore as follows, according to physicists:
$$\begin{aligned}
@@ -51,7 +51,7 @@ $$\begin{gathered}
H_4(x) = 16 x^4 - 48 x^2 + 12
\end{gathered}$$
-And then more $H_n$ can be computed quickly
+And then more $$H_n$$ can be computed quickly
using the following recurrence relation:
$$\begin{aligned}
@@ -70,7 +70,7 @@ $$\begin{aligned}
}
\end{aligned}$$
-Importantly, all $H_n$ are orthogonal with respect to the weight function $w(x) \equiv \exp(- x^2)$:
+Importantly, all $$H_n$$ are orthogonal with respect to the weight function $$w(x) \equiv \exp(- x^2)$$:
$$\begin{aligned}
\boxed{
@@ -80,10 +80,10 @@ $$\begin{aligned}
}
\end{aligned}$$
-Where $\delta_{nm}$ is the Kronecker delta.
+Where $$\delta_{nm}$$ is the Kronecker delta.
Finally, 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.
-This means that every such $f$ can be expanded in $H_n$:
+of all functions $$f(x)$$ for which $$\Inprod{f}{w f}$$ is finite.
+This means that every such $$f$$ can be expanded in $$H_n$$:
$$\begin{aligned}
\boxed{