Editing Legendre polynomials (section)

==Applications==

===Expanding an inverse distance potential===
{{main|Laplace expansion (potential)}}

The Legendre polynomials were first introduced in 1782 by [[Adrien-Marie Legendre]]<ref>{{cite book |first1=A.-M. |last1=Legendre |chapter=Recherches sur l'attraction des sphéroïdes homogènes |title=Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées |volume=X |pages=411–435 |location=Paris |date=1785 |orig-year=1782 |language=fr |chapter-url=http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf |url-status=dead |archive-url=https://web.archive.org/web/20090920070434/http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf |archive-date=2009-09-20 }}</ref> as the coefficients in the expansion of the [[Newtonian potential]]
<math display="block">\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma),</math>
where {{math|''r''}} and {{math|''r''′}} are the lengths of the vectors {{math|'''x'''}} and {{math|'''x'''′}} respectively and {{math|''γ''}} is the angle between those two vectors. The series converges when {{math|''r'' > ''r''′}}. The expression gives the [[gravitational potential]] associated to a [[point mass]] or the [[Coulomb potential]] associated to a [[point charge]]. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.

Legendre polynomials occur in the solution of [[Laplace's equation]] of the static [[electric potential|potential]], {{math|1=∇<sup>2</sup> Φ('''x''') = 0}}, in a charge-free region of space, using the method of [[separation of variables]], where the [[boundary conditions]] have axial symmetry (no dependence on an [[azimuth|azimuthal angle]]). Where {{math|'''ẑ'''}} is the axis of symmetry and {{math|''θ''}} is the angle between the position of the observer and the {{math|'''ẑ'''}} axis (the zenith angle), the solution for the potential will be
<math display="block">\Phi(r,\theta) = \sum_{\ell=0}^\infty \left( A_\ell r^\ell + B_\ell r^{-(\ell+1)} \right) P_\ell(\cos\theta) \,.</math>

{{math|''A<sub>l</sub>''}} and {{math|''B<sub>l</sub>''}} are to be determined according to the boundary condition of each problem.<ref>{{cite book|last=Jackson |first=J. D. |title=Classical Electrodynamics |url=https://archive.org/details/classicalelectro00jack_449 |url-access=limited |edition= 3rd |location=Wiley & Sons |date=1999 |page=[https://archive.org/details/classicalelectro00jack_449/page/n102 103] |isbn=978-0-471-30932-1}}</ref>

They also appear when solving the [[Schrödinger equation]] in three dimensions for a central force.

=== In multipole expansions ===
[[File:Point axial multipole.svg|right|Diagram for the multipole expansion of electric potential.]]
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
<math display="block">\frac{1}{\sqrt{1 + \eta^2 - 2\eta x}} = \sum_{k=0}^\infty \eta^k P_k(x),</math>
which arise naturally in [[multipole expansion]]s.  The left-hand side of the equation is the [[generating function]] for the Legendre polynomials.

As an example, the [[electric potential]] {{math|Φ(''r'',''θ'')}} (in [[spherical coordinates]]) due to a [[point charge]] located on the {{math|''z''}}-axis at {{math|1=''z'' = ''a''}} (see diagram right) varies as
<math display="block">\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^2 + a^2 - 2ar \cos\theta}}.</math>

If the radius {{math|''r''}} of the observation point {{math|P}} is greater than {{math|''a''}}, the potential may be expanded in the Legendre polynomials
<math display="block">\Phi(r, \theta) \propto \frac{1}{r} \sum_{k=0}^\infty \left( \frac{a}{r} \right)^k P_k(\cos \theta),</math>
where we have defined {{math|1=''η'' = {{sfrac|''a''|''r''}} < 1}} and {{math|1=''x'' = cos ''θ''}}. This expansion is used to develop the normal [[multipole expansion]].

Conversely, if the radius {{math|''r''}} of the observation point {{math|P}} is smaller than {{math|''a''}}, the potential may still be expanded in the Legendre polynomials as above, but with {{math|''a''}} and {{math|''r''}} exchanged. This expansion is the basis of [[interior multipole expansion]].

=== In trigonometry ===

The trigonometric functions {{math|cos ''nθ''}}, also denoted as the [[Chebyshev polynomials]] {{math|''T<sub>n</sub>''(cos ''θ'') ≡ cos ''nθ''}}, can also be multipole expanded by the Legendre polynomials {{math|''P<sub>n</sub>''(cos ''θ'')}}. The first several orders are as follows:
<math display="block">\begin{alignat}{2}
T_0(\cos\theta)&=1           &&=P_0(\cos\theta),\\[4pt]
T_1(\cos\theta)&=\cos  \theta&&=P_1(\cos\theta),\\[4pt]
T_2(\cos\theta)&=\cos 2\theta&&=\tfrac{1}{3}\bigl(4P_2(\cos\theta)-P_0(\cos\theta)\bigr),\\[4pt]
T_3(\cos\theta)&=\cos 3\theta&&=\tfrac{1}{5}\bigl(8P_3(\cos\theta)-3P_1(\cos\theta)\bigr),\\[4pt]
T_4(\cos\theta)&=\cos 4\theta&&=\tfrac{1}{105}\bigl(192P_4(\cos\theta)-80P_2(\cos\theta)-7P_0(\cos\theta)\bigr),\\[4pt]
T_5(\cos\theta)&=\cos 5\theta&&=\tfrac{1}{63}\bigl(128P_5(\cos\theta)-56P_3(\cos\theta)-9P_1(\cos\theta)\bigr),\\[4pt]
T_6(\cos\theta)&=\cos 6\theta&&=\tfrac{1}{1155}\bigl(2560P_6(\cos\theta)-1152P_4(\cos\theta)-220P_2(\cos\theta)-33P_0(\cos\theta)\bigr).
\end{alignat}</math>

This can be summarized for <math>n>0</math> as

<math>
T_n(x)=2^{2n-n'}\hat n!\sum_{t=0}^{\hat n} (n-2t+1/2)
\frac{(n-t-1)!}{2^{2t}t!(n-1)!}
\times
\frac{(-1)\cdot 1\cdot 3\cdots (2t-3)}{(1+2n')(3+2n')\cdots (2n-2t+1)}P_{n-2t}(x)
.
</math>

where <math>\hat n\equiv \lfloor n/2\rfloor</math>,
<math>n'\equiv \lfloor (n+1)/2\rfloor</math>, and where the products
with the steps of two in the numerator and denominator are 
to be interpreted as 1 if the are empty, i.e., if the last factor is smaller than the
first factor.


Another property is the expression for {{math|sin (''n'' + 1)''θ''}}, which is
<math display="block">\frac{\sin (n+1)\theta}{\sin\theta}=\sum_{\ell=0}^n P_\ell(\cos\theta) P_{n-\ell}(\cos\theta).</math>

=== In recurrent neural networks ===

A [[recurrent neural network]] that contains a {{math|''d''}}-dimensional memory vector, <math>\mathbf{m} \in \R^d</math>, can be optimized such that its neural activities obey the [[linear time-invariant system]] given by the following [[state-space representation]]:
<math display="block">\theta \dot{\mathbf{m}}(t) = A\mathbf{m}(t) + Bu(t),</math>
<math display="block">\begin{align}
A &= \left[ a \right]_{ij} \in \R^{d \times d} \text{,} \quad
&& a_{ij} = \left(2i + 1\right)
\begin{cases}
  -1 & i < j \\
  (-1)^{i-j+1} & i \ge j
\end{cases},\\
B &= \left[ b \right]_i \in \R^{d \times 1} \text{,} \quad
&& b_i = (2i + 1) (-1)^i .
\end{align}</math>

In this case, the sliding window of <math>u</math> across the past <math>\theta</math> units of time is [[Approximation theory|best approximated]] by a linear combination of the first <math>d</math> shifted Legendre polynomials, weighted together by the elements of <math>\mathbf{m}</math> at time <math>t</math>:
<math display="block">u(t - \theta') \approx \sum_{\ell=0}^{d-1} \widetilde{P}_\ell \left(\frac{\theta'}{\theta} \right) \, m_{\ell}(t) , \quad 0 \le \theta' \le \theta .</math>

When combined with [[deep learning]] methods, these networks can be trained to outperform [[long short-term memory]] units and related architectures, while using fewer computational resources.<ref>{{cite conference |last1=Voelker |first1=Aaron R. |last2=Kajić |first2=Ivana |last3=Eliasmith |first3=Chris |title=Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks |url=http://compneuro.uwaterloo.ca/files/publications/voelker.2019.lmu.pdf |conference=Advances in Neural Information Processing Systems |conference-url=https://neurips.cc |year=2019 }}</ref>