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diff --git a/source/know/concept/laguerre-polynomials/index.md b/source/know/concept/laguerre-polynomials/index.md new file mode 100644 index 0000000..130dff2 --- /dev/null +++ b/source/know/concept/laguerre-polynomials/index.md @@ -0,0 +1,125 @@ +--- +title: "Laguerre polynomials" +date: 2021-09-08 +categories: +- Mathematics +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: + +$$\begin{aligned} + \boxed{ + x u'' + (1 - x) u' + n u = 0 + } +\end{aligned}$$ + +The $n$th-order Laguerre polynomial $L_n(x)$ +is given in the form of a *Rodrigues' formula* by: + +$$\begin{aligned} + L_n(x) + &= \frac{1}{n!} \exp(x) \dvn{n}{}{x}\big(x^n \exp(-x)\big) + \\ + &= \frac{1}{n!} \Big( \dv{}{x}- 1 \Big)^n x^n +\end{aligned}$$ + +The first couple of Laguerre polynomials $L_n(x)$ are therefore as follows: + +$$\begin{gathered} + L_0(x) = 1 + \qquad \quad + L_1(x) = 1 - x + \qquad \quad + L_2(x) = \frac{1}{2} (x^2 - 4 x + 2) +\end{gathered}$$ + +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: + +$$\begin{aligned} + \boxed{ + x u'' + (\alpha + 1 - x) u' + n u = 0 + } +\end{aligned}$$ + +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 $L_n^\alpha = L_n$: + +$$\begin{aligned} + L_n^\alpha(x) + &= \frac{1}{n!} x^{-\alpha} \exp(x) \dvn{n}{}{x}\big( x^{n + \alpha} \exp(-x) \big) + \\ + &= \frac{x^{-\alpha}}{n!} \Big( \dv{}{x}- 1 \Big)^n x^{n + \alpha} +\end{aligned}$$ + +The first couple of associated Laguerre polynomials $L_n^\alpha(x)$ are therefore as follows: + +$$\begin{aligned} + L_0^\alpha(x) = 1 + \qquad + L_1^\alpha(x) = \alpha + 1 - x + \qquad + L_2^\alpha(x) = \frac{1}{2} (x^2 - 2 \alpha x - 4 x + \alpha^2 + 3 \alpha + 2) +\end{aligned}$$ + +And then more $L_n^\alpha$ can be computed quickly +using the following recurrence relation: + +$$\begin{aligned} + \boxed{ + L_{n + 1}^\alpha(x) + = \frac{(\alpha + 2 n + 1 - x) L_n^\alpha(x) - (\alpha + n) L_{n - 1}^\alpha(x)}{n + 1} + } +\end{aligned}$$ + +The derivatives are also straightforward to calculate +using the following relation: + +$$\begin{aligned} + \boxed{ + \dvn{k}{}{x}L_n^\alpha(x) + = (-1)^k L_{n - k}^{\alpha + k}(x) + } +\end{aligned}$$ + +Noteworthy is that these polynomials (both normal and associated) +are all mutually orthogonal for $x \in [0, \infty[$, +with respect to the weight function $w(x) \equiv x^\alpha \exp(-x)$: + +$$\begin{aligned} + \boxed{ + \Inprod{L_m^\alpha}{w L_n^\alpha} + = \int_0^\infty L_m^\alpha(x) \: L_n^\alpha(x) \: w(x) \dd{x} + = \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{nm} + } +\end{aligned}$$ + +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: + +$$\begin{aligned} + \boxed{ + f(x) + = \sum_{n = 0}^\infty a_n L_n^\alpha(x) + = \sum_{n = 0}^\infty \frac{\Inprod{L_n}{w f}}{\Inprod{L_n}{w L_n}} L_n^\alpha(x) + } +\end{aligned}$$ + +Finally, the $L_n^\alpha(x)$ are related to +the [Hermite polynomials](/know/concept/hermite-polynomials/) $H_n(x)$ like so: + +$$\begin{aligned} + H_{2n(x)} &= (-1)^n 2^{2n} n! \: L_n^{-1/2}(x^2) + \\ + H_{2n + 1(x)} &= (-1)^n 2^{2n + 1} n! \: L_n^{1/2}(x^2) +\end{aligned}$$ |