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Diffstat (limited to 'source/know/concept/legendre-polynomials')
| -rw-r--r-- | source/know/concept/legendre-polynomials/index.md | 38 |
1 files changed, 19 insertions, 19 deletions
diff --git a/source/know/concept/legendre-polynomials/index.md b/source/know/concept/legendre-polynomials/index.md index 74543a3..f223cd3 100644 --- a/source/know/concept/legendre-polynomials/index.md +++ b/source/know/concept/legendre-polynomials/index.md @@ -8,10 +8,10 @@ layout: "concept" --- The **Legendre polynomials** are a set of functions that sometimes arise in physics. -They are the eigenfunctions $u(x)$ of **Legendre's differential equation**, +They are the eigenfunctions $$u(x)$$ of **Legendre's differential equation**, which is a ([Sturm-Liouville](/know/concept/sturm-liouville-theory/)) -eigenvalue problem for $\ell (\ell + 1)$, -where $\ell$ turns out to be a non-negative integer: +eigenvalue problem for $$\ell (\ell + 1)$$, +where $$\ell$$ turns out to be a non-negative integer: $$\begin{aligned} \boxed{ @@ -19,7 +19,7 @@ $$\begin{aligned} } \end{aligned}$$ -The $\ell$th-degree Legendre polynomial $P_\ell(x)$ +The $$\ell$$th-degree Legendre polynomial $$P_\ell(x)$$ is given in the form of a *Rodrigues' formula* by: $$\begin{aligned} @@ -27,7 +27,7 @@ $$\begin{aligned} &= \frac{1}{2^\ell \ell!} \dvn{\ell}{}{x}(x^2 - 1)^\ell \end{aligned}$$ -The first handful of Legendre polynomials $P_\ell(x)$ are therefore as follows: +The first handful of Legendre polynomials $$P_\ell(x)$$ are therefore as follows: $$\begin{gathered} P_0(x) = 1 @@ -41,7 +41,7 @@ $$\begin{gathered} P_4(x) = \frac{1}{8} (35 x^4 - 30 x^2 + 3) \end{gathered}$$ -And then more $P_\ell$ can be computed quickly +And then more $$P_\ell$$ can be computed quickly using **Bonnet's recursion formula**: $$\begin{aligned} @@ -50,7 +50,7 @@ $$\begin{aligned} } \end{aligned}$$ -The derivative of a given $P_\ell$ can be calculated recursively +The derivative of a given $$P_\ell$$ can be calculated recursively using the following relation: $$\begin{aligned} @@ -61,7 +61,7 @@ $$\begin{aligned} \end{aligned}$$ Noteworthy is that the Legendre polynomials -are mutually orthogonal for $x \in [-1, 1]$: +are mutually orthogonal for $$x \in [-1, 1]$$: $$\begin{aligned} \boxed{ @@ -74,7 +74,7 @@ $$\begin{aligned} As was to be expected from Sturm-Liouville theory. Likewise, they form a complete basis in the [Hilbert space](/know/concept/hilbert-space/) -of piecewise continuous functions $f(x)$ on $x \in [-1, 1]$, +of piecewise continuous functions $$f(x)$$ on $$x \in [-1, 1]$$, meaning: $$\begin{aligned} @@ -85,11 +85,11 @@ $$\begin{aligned} } \end{aligned}$$ -Each Legendre polynomial $P_\ell$ comes with -a set of **associated Legendre polynomials** $P_\ell^m(x)$ -of order $m$ and degree $\ell$. +Each Legendre polynomial $$P_\ell$$ comes with +a set of **associated Legendre polynomials** $$P_\ell^m(x)$$ +of order $$m$$ and degree $$\ell$$. These are the non-singular solutions of the **general Legendre equation**, -where $m$ and $\ell$ are integers satisfying $-\ell \le m \le \ell$: +where $$m$$ and $$\ell$$ are integers satisfying $$-\ell \le m \le \ell$$: $$\begin{aligned} \boxed{ @@ -97,17 +97,17 @@ $$\begin{aligned} } \end{aligned}$$ -The $\ell$th-degree $m$th-order associated Legendre polynomial $P_\ell^m$ -is as follows for $m \ge 0$: +The $$\ell$$th-degree $$m$$th-order associated Legendre polynomial $$P_\ell^m$$ +is as follows for $$m \ge 0$$: $$\begin{aligned} P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \dvn{m}{}{x}P_\ell(x) \end{aligned}$$ -Here, the $(-1)^m$ in front is called the **Condon-Shortley phase**, +Here, the $$(-1)^m$$ in front is called the **Condon-Shortley phase**, and is omitted by some authors. -For negative orders $m$, +For negative orders $$m$$, an additional constant factor is necessary: $$\begin{aligned} @@ -115,6 +115,6 @@ $$\begin{aligned} \end{aligned}$$ Beware, the name is misleading: -if $m$ is odd, then $P_\ell^m$ is actually not a polynomial. -Moreover, not all $P_\ell^m$ are mutually orthogonal +if $$m$$ is odd, then $$P_\ell^m$$ is actually not a polynomial. +Moreover, not all $$P_\ell^m$$ are mutually orthogonal (but some are). |