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Diffstat (limited to 'source/know/concept/hermite-polynomials')
| -rw-r--r-- | source/know/concept/hermite-polynomials/index.md | 20 |
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{ |