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Diffstat (limited to 'source/know/concept/laguerre-polynomials')
| -rw-r--r-- | source/know/concept/laguerre-polynomials/index.md | 34 |
1 files changed, 17 insertions, 17 deletions
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) |