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{{for|Legendre's Diophantine equation|Legendre's equation}}
{{selfref|[[Associated Legendre polynomials]] are the most general solution to Legendre's Equation and '''Legendre polynomials''' are solutions that are azimuthally symmetric.}}
In [[mathematics]], '''Legendre  functions''' are solutions to '''Legendre's differential equation''':
 
:<math>{d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0.</math>
 
They are named after [[Adrien-Marie Legendre]]. This [[differential equation|ordinary differential equation]] is frequently encountered in [[physics]] and other technical fields. In particular, it occurs when solving [[Laplace's equation]] (and related [[partial differential equation]]s) in [[spherical coordinates]]. 
 
The Legendre differential equation may be solved using the standard [[power series]] method. The equation has [[regular singular point]]s at ''x''&nbsp;=&nbsp;±1 so, in general, a series solution about the origin will only converge for |''x''|&nbsp;&lt;&nbsp;1. When ''n'' is an integer, the solution ''P''<sub>''n''</sub>(''x'') that is regular at ''x''&nbsp;=&nbsp;1 is also regular at ''x''&nbsp;=&nbsp;&minus;1, and the series for this solution terminates (i.e. it is a polynomial).
 
These solutions for ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;... (with the normalization ''P<sub>n</sub>''(1)&nbsp;=&nbsp;1) form a [[polynomial sequence]] of [[orthogonal polynomials]] called the '''Legendre polynomials'''. Each Legendre polynomial ''P''<sub>''n''</sub>(''x'') is an ''n''th-degree polynomial. It may be expressed using [[Rodrigues' formula]]:
 
:<math>P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right]. </math>
 
That these polynomials satisfy the Legendre differential equation ({{EquationNote|1}}) follows by differentiating (''n''+1) times both sides of the identity
 
:<math>(x^2-1)\frac{d}{dx}(x^2-1)^n = 2nx(x^2-1)^n</math>
 
and employing the [[general Leibniz rule]] for repeated differentiation.<ref>{{harvnb|Courant|Hilbert|1953|loc=II, §8}}</ref>  The ''P''<sub>''n''</sub> can also be defined as the coefficients in a [[Taylor series]] expansion:<ref name="arfken">{{Citation |author=[[George B. Arfken]], Hans J. Weber |title=Mathematical Methods for Physicists |publisher=Elsevier Academic Press |year=2005 |page=743 |isbn=0-12-059876-0}}</ref>
 
:<math>\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n.\qquad (1)</math>
 
In physics, this [[generating function]] is the basis for [[multipole expansion]]s.
 
== Recursive definition ==
Expanding the Taylor series in equation (1) for the first two terms gives
:<math>P_0(x) = 1,\quad P_1(x) = x</math>
for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (1) is differentiated with respect to t on both sides and rearranged to obtain
:<math>\frac{x-t}{\sqrt{1-2xt+t^2}} = (1-2xt+t^2) \sum_{n=1}^\infty n P_n(x) t^{n-1}.</math>
Replacing the quotient of the square root with its definition in (1), and [[equating the coefficients]] of powers of t in the resulting expansion gives ''Bonnet’s recursion formula''
:<math> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x).\,</math>
This relation, along with the first two polynomials ''P''<sub>0</sub> and ''P''<sub>1</sub>, allows the Legendre Polynomials to be generated recursively.
 
Explicit representations include
:<math>P_n(x)= \frac 1 {2^n} \sum_{k=0}^n {n\choose k}^2 (x-1)^{n-k}(x+1)^k=\sum_{k=0}^n {n\choose k} {-n-1\choose k} \left( \frac{1-x}{2} \right)^k= 2^n\cdot \sum_{k=0}^n x^k {n \choose k}{\frac{n+k-1}2\choose n},</math>
where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the [[binomial coefficient]].
 
The first few Legendre polynomials are:
<center><table style="background:white;">
<tr>
<td width="20%" align="center">'''n'''</td>
<td align="center"><math>P_n(x)\,</math></td>
</tr>
<tr>
<td align="center">0</td>
<td align="center"><math>1\,</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="center"><math>x\,</math></td>
</tr>
<tr>
<td align="center">2</td>
<td align="center"><math>\begin{matrix}\frac12\end{matrix} (3x^2-1) \,</math></td>
</tr>
<tr>
<td align="center">3</td>
<td align="center"><math>\begin{matrix}\frac12\end{matrix} (5x^3-3x) \,</math></td>
</tr>
<tr>
<td align="center">4</td>
<td align="center"><math>\begin{matrix}\frac18\end{matrix} (35x^4-30x^2+3)\,</math>
</tr>
<tr>
<td align="center">5</td>
<td align="center"><math>\begin{matrix}\frac18\end{matrix} (63x^5-70x^3+15x)\,</math>
</tr>
<tr>
<td align="center">6</td>
<td align="center"><math>\begin{matrix}\frac1{16}\end{matrix} (231x^6-315x^4+105x^2-5)\,</math>
</tr>
<tr>
<td align="center">7</td>
<td align="center"><math>\begin{matrix}\frac1{16}\end{matrix} (429x^7-693x^5+315x^3-35x)\,</math>
</tr>
<tr>
<td align="center">8</td>
<td align="center"><math>\begin{matrix}\frac1{128}\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\,</math>
</tr>
<tr>
<td align="center">9</td>
<td align="center"><math>\begin{matrix}\frac1{128}\end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x)\,</math>
</tr>
<tr>
<td align="center">10</td>
<td align="center"><math>\begin{matrix}\frac1{256}\end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)\,</math>
</tr>
</table>
</center>
 
The graphs of these polynomials (up to ''n'' = 5) are shown below:
[[File:Legendrepolynomials6.svg|640px|center]]
 
== Orthogonality ==
 
An important property of the Legendre polynomials is that they are [[orthogonal]] with respect to the [[Lp space|L<sup>2</sup> inner product]] on the interval &minus;1&nbsp;≤&nbsp;''x''&nbsp;≤&nbsp;1:
 
:<math>\int_{-1}^{1} P_m(x) P_n(x)\,dx = {2 \over {2n + 1}} \delta_{mn}</math>
 
(where δ<sub>''mn''</sub> denotes the [[Kronecker delta]], equal to 1 if ''m''&nbsp;=&nbsp;''n'' and to 0 otherwise).
In fact, an alternative derivation of the Legendre polynomials is by carrying out the [[Gram-Schmidt process]] on the polynomials {1,&nbsp;''x'',&nbsp;''x''<sup>2</sup>,&nbsp;...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a [[Sturm&ndash;Liouville theory|Sturm–Liouville problem]], where the Legendre polynomials are [[eigenfunction]]s of a [[hermitian operator|Hermitian]] [[differential operator]]:
 
:<math>{d \over dx} \left[ (1-x^2) {d \over dx} P(x) \right] = -\lambda P(x),</math>
 
where the eigenvalue λ corresponds to ''n''(''n''&nbsp;+&nbsp;1).
 
==Applications of Legendre polynomials in physics==
 
The Legendre polynomials were first introduced in 1782 by [[Adrien-Marie Legendre]]<ref>M. Le Gendre, "Recherches sur l'attraction des sphéroïdes homogènes," ''Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées'', Tome X, pp. 411-435 (Paris, 1785).  [Note:  Legendre submitted his findings to the Academy in 1782, but they were published in 1785.]  Available on-line (in French) at:  http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf  .</ref> as the coefficients in the expansion of the [[Newtonian potential]]
:<math>
\frac{1}{\left| \mathbf{x}-\mathbf{x}^\prime \right|} = \frac{1}{\sqrt{r^2+r^{\prime 2}-2rr'\cos\gamma}} = \sum_{\ell=0}^{\infty} \frac{r^{\prime \ell}}{r^{\ell+1}} P_{\ell}(\cos \gamma)
</math>
 
where <math>r</math> and <math>r'</math> are the lengths of the vectors <math>\mathbf{x}</math> and <math>\mathbf{x}^\prime</math> respectively and <math>\gamma</math> is the angle between those two vectors. The series converges when <math>r>r'</math>.  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 equation]] of the [[electric potential|potential]], <math>\nabla^2 \Phi(\mathbf{x})=0</math>, 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>\widehat{\mathbf{z}}</math> is the axis of symmetry and <math>\theta</math> is the angle between the position of the observer and the <math>\widehat{\mathbf{z}}</math> axis (the zenith angle), the solution for the potential will be
 
:<math>
\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_\ell</math> and <math>B_\ell</math> are to be determined according to the boundary condition of each problem.<ref>Jackson, J.D. ''Classical Electrodynamics'', 3rd edition, Wiley & Sons, 1999. page 103</ref>
 
They also appear when solving Schrödinger equation in three dimensions for a central force.
 
'''Legendre polynomials in multipole expansions'''
[[File:Point axial multipole.svg|frame|right|Figure 2]]
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
:<math>
\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>\Phi(r, \theta)</math> (in [[spherical coordinates]])  due to a [[point charge]] located on the ''z''-axis at  <math>z=a</math> (Figure&nbsp;2) varies like
 
:<math>
\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^{2} + a^{2} - 2ar \cos\theta}}.
</math>
 
If the radius ''r'' of the observation point '''P''' is
greater than ''a'', the potential may be expanded in the Legendre polynomials
 
:<math>
\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 ''η''&nbsp;=&nbsp;''a''/''r''&nbsp;<&nbsp;1 and ''x''&nbsp;=&nbsp;cos&nbsp;''θ''.  This expansion is used to develop the normal [[multipole expansion]].
 
Conversely, if the radius ''r'' of the observation point '''P''' is
smaller than ''a'', the potential may still be expanded in the
Legendre polynomials as above, but with ''a'' and ''r'' exchanged.
This expansion is the basis of [[interior multipole expansion]].
 
== Additional properties of Legendre polynomials ==
 
Legendre polynomials are symmetric or antisymmetric, that is
 
:<math>P_n(-x) = (-1)^n P_n(x). \,</math><ref name="arfken">George B. Arfken, Hans J. Weber, ''Mathematical Methods for Physicists'', Elsevier Academic Press, 2005, pg. 753.</ref>
 
Since the differential equation and the orthogonality property are
independent of scaling, the Legendre polynomials' definitions are
"standardized" (sometimes called "normalization", but note that the
actual norm is not unity) by being scaled so that
 
:<math>P_n(1) = 1. \,</math>
 
The derivative at the end point is given by
 
:<math>P_n'(1) = \frac{n(n+1)}{2}. \, </math>
 
As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula
:<math> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)\,</math>
and
:<math> {x^2-1 \over n} {d \over dx} P_n(x) = xP_n(x) - P_{n-1}(x).</math>
 
Useful for the integration of Legendre polynomials is
 
:<math>(2n+1) P_n(x) = {d \over dx} \left[ P_{n+1}(x) - P_{n-1}(x) \right].</math>
 
From the above one can see also that
 
:<math>{d \over dx} P_{n+1}(x) = (2n+1) P_n(x) + (2(n-2)+1) P_{n-2}(x) + (2(n-4)+1) P_{n-4}(x) + \ldots</math>
 
or equivalently
 
:<math>{d \over dx} P_{n+1}(x) = {2 P_n(x) \over \| P_n(x) \|^2} + {2 P_{n-2}(x) \over \| P_{n-2}(x) \|^2}+\ldots</math>
 
where <math>\| P_n(x) \|</math> is the norm over the interval −1 ≤ x ≤ 1
 
:<math>\| P_n(x) \| = \sqrt{\int _{- 1}^{1}(P_n(x))^2 \,dx} = \sqrt{\frac{2}{2 n + 1}}.</math>
 
From Bonnet’s recursion formula one obtains by induction the explicit representation
:<math>P_n(x) = \sum_{k=0}^n (-1)^k \begin{pmatrix} n \\ k \end{pmatrix}^2 \left( \frac{1+x}{2} \right)^{n-k} \left( \frac{1-x}{2} \right)^k.</math>
 
The [[Askey–Gasper inequality]] for Legendre polynomials reads
:<math>\sum_{j=0}^n P_j(x)\ge 0\qquad (x\ge -1).</math>
 
== Shifted Legendre polynomials ==
 
The '''shifted Legendre polynomials''' are defined as <math>\tilde{P_n}(x) = P_n(2x-1)</math>. Here the "shifting" function <math>x\mapsto 2x-1</math> (in fact, it is an [[affine transformation]]) is chosen such that it [[bijection|bijectively maps]] the interval [0,&nbsp;1] to the interval [&minus;1,&nbsp;1], implying that the polynomials <math>\tilde{P_n}(x)</math> are orthogonal on [0,&nbsp;1]:
 
:<math>\int_{0}^{1} \tilde{P_m}(x) \tilde{P_n}(x)\,dx = {1 \over {2n + 1}} \delta_{mn}.</math>
 
An explicit expression for the shifted Legendre polynomials is given by
 
:<math>\tilde{P_n}(x) = (-1)^n \sum_{k=0}^n {n \choose k} {n+k \choose k} (-x)^k.</math>
 
The analogue of [[Rodrigues' formula]] for the shifted Legendre polynomials is
 
:<math>\tilde{P_n}(x) = \frac{1}{n!} {d^n \over dx^n } \left[ (x^2 -x)^n \right].\, </math>
 
The first few shifted Legendre polynomials are:
 
<center>
{| class="wikitable"
|'''''n'''''
| align=center | <math>\tilde{P_n}(x)</math>
|-
| 0
| 1
|-
| 1
| <math>2x-1</math>
|-
| 2
| <math>6x^2-6x+1</math>
|-
| 3
| <math>20x^3-30x^2+12x-1</math>
|-
|4
| <math>70x^4-140x^3+90x^2-20x+1</math>
|}
</center>
 
==Legendre functions==
 
As well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series.  These are the ''Legendre functions of the second kind'', denoted by <math>Q_n(x)</math>.
:<math>Q_n(x)=\frac{n!}{1.3\cdots(2n+1)}\left[x^{-(n+1)}+\frac{(n+1)(n+2)}{2(n+3)}x^{-(n+3)}+\frac{(n+1)(n+2)(n+3)(n+4)}{2.4(2n+3)(2n+5)}x^{-(n+5)}+\cdots\right]</math>
 
The differential equation
 
:<math>{d \over dx} \left[ (1-x^2) {d \over dx} f(x) \right] + n(n+1)f(x) = 0</math>
has the general solution
:<math>f(x)=AP_n(x)+BQ_n(x)</math>,
where ''A'' and ''B'' are constants.
 
== Legendre functions of fractional order ==
{{main|Legendre function}}
Legendre functions of fractional order exist and follow from insertion of fractional derivatives as defined by [[fractional calculus]] and non-integer [[factorial]]s (defined by the [[gamma function]]) into the [[Rodrigues' formula]]. The resulting functions continue to satisfy the Legendre differential equation throughout (&minus;1,1), but are no longer regular at the endpoints.  The fractional order Legendre function ''P''<sub>''n''</sub> agrees with the [[associated Legendre polynomial]] ''P''{{su|b=''n''|p=0}}.
 
==See also==
* [[Associated Legendre function]]s
* [[Gaussian quadrature]]
* [[Gegenbauer polynomials]]
* [[Legendre rational functions]]
* [[Turán's inequalities]]
* [[Legendre wavelet]]
* [[Jacobi polynomials]]
* [[Spherical Harmonics]]
 
== Notes ==
{{Reflist}}
 
== References ==
* {{Abramowitz_Stegun_ref2|8|332|22|773}}
* {{citation|last=Bayin|first=S.S.|year=2006|title=Mathematical Methods in Science and Engineering|publisher=Wiley}}, Chapter 2.
* {{citation|last=Belousov|first=S. L.|year=1962|title=Tables of normalized associated Legendre polynomials|series=Mathematical tables|volume=18|publisher=Pergamon Press}}.
* {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publischer, Inc|publication-place=New York}}.
*{{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
*{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|authorlink=Tom H. Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René  F. |last4=Swarttouw}}
* {{Citation | author=Refaat El Attar | title= Legendre Polynomials and Functions | publisher= CreateSpace | year=2009 | isbn = 978-1-4414-9012-4}}
 
==External links==
*[http://www.physics.drexel.edu/~tim/open/hydrofin A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen]
*{{springer|title=Legendre polynomials|id=p/l058050}}
*[http://mathworld.wolfram.com/LegendrePolynomial.html Wolfram MathWorld entry on Legendre polynomials]
*[http://math.fullerton.edu/mathews/n2003/LegendrePolyMod.html Module for Legendre Polynomials by John H. Mathews]
*[http://www.du.edu/~jcalvert/math/legendre.htm Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics]
*[http://www.morehouse.edu/facstaff/cmoore/Legendre%20Polynomials.htm The Legendre Polynomials by Carlyle E. Moore]
*[http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html Legendre Polynomials from Hyperphysics]
 
{{DEFAULTSORT:Legendre Polynomials}}
[[Category:Special hypergeometric functions]]
[[Category:Orthogonal polynomials]]
[[Category:Polynomials]]

Revision as of 09:59, 27 November 2013

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Making the leap to turning into a property investor We 'assumed' we'd just contact an estate agent and they'd have a list of properties for hire, if we didn't like what they had we would then transfer on to a different agent and keep doing this till we positioned an condominium that was suitable for us. Stamp obligation is to be paid inside 14 days from the date of acceptance of the OTP or Sale and Buy (S&P) Agreement. For extra data, please go to www.iras.gov.sg Estaland Property Associates Pte Ltd Completely different agents may bring the same buyer to view your house a second time, leading to confusion. Brokers who don't have exclusive rights to an inventory could also be much less keen to work for that itemizing. including Estate Agents Act and regulation of real property industry) In mathematics, Legendre functions are solutions to Legendre's differential equation:

They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates.

The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is an integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial).

These solutions for n = 0, 1, 2, ... (with the normalization Pn(1) = 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:

That these polynomials satisfy the Legendre differential equation (Template:EquationNote) follows by differentiating (n+1) times both sides of the identity

and employing the general Leibniz rule for repeated differentiation.[1] The Pn can also be defined as the coefficients in a Taylor series expansion:[2]

In physics, this generating function is the basis for multipole expansions.

Recursive definition

Expanding the Taylor series in equation (1) for the first two terms gives

for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (1) is differentiated with respect to t on both sides and rearranged to obtain

Replacing the quotient of the square root with its definition in (1), and equating the coefficients of powers of t in the resulting expansion gives Bonnet’s recursion formula

This relation, along with the first two polynomials P0 and P1, allows the Legendre Polynomials to be generated recursively.

Explicit representations include

where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient.

The first few Legendre polynomials are:

n
0
1
2
3
4
5
6
7
8
9
10

The graphs of these polynomials (up to n = 5) are shown below:

Orthogonality

An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval −1 ≤ x ≤ 1:

(where δmn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise). In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1, xx2, ...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a Sturm–Liouville problem, where the Legendre polynomials are eigenfunctions of a Hermitian differential operator:

where the eigenvalue λ corresponds to n(n + 1).

Applications of Legendre polynomials in physics

The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre[3] as the coefficients in the expansion of the Newtonian potential

where and are the lengths of the vectors and respectively and is the angle between those two vectors. The series converges when . 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 equation of the potential, , 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 azimuthal angle). Where is the axis of symmetry and is the angle between the position of the observer and the axis (the zenith angle), the solution for the potential will be

and are to be determined according to the boundary condition of each problem.[4]

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

Legendre polynomials in multipole expansions

Figure 2

Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):

which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.

As an example, the electric potential (in spherical coordinates) due to a point charge located on the z-axis at (Figure 2) varies like

If the radius r of the observation point P is greater than a, the potential may be expanded in the Legendre polynomials

where we have defined η = a/r < 1 and x = cos θ. This expansion is used to develop the normal multipole expansion.

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

Additional properties of Legendre polynomials

Legendre polynomials are symmetric or antisymmetric, that is

[2]

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not unity) by being scaled so that

The derivative at the end point is given by

As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula

and

Useful for the integration of Legendre polynomials is

From the above one can see also that

or equivalently

where is the norm over the interval −1 ≤ x ≤ 1

From Bonnet’s recursion formula one obtains by induction the explicit representation

The Askey–Gasper inequality for Legendre polynomials reads

Shifted Legendre polynomials

The shifted Legendre polynomials are defined as . Here the "shifting" function (in fact, it is an affine transformation) is chosen such that it bijectively maps the interval [0, 1] to the interval [−1, 1], implying that the polynomials are orthogonal on [0, 1]:

An explicit expression for the shifted Legendre polynomials is given by

The analogue of Rodrigues' formula for the shifted Legendre polynomials is

The first few shifted Legendre polynomials are:

n
0 1
1
2
3
4

Legendre functions

As well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series. These are the Legendre functions of the second kind, denoted by .

The differential equation

has the general solution

,

where A and B are constants.

Legendre functions of fractional order

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Legendre functions of fractional order exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula. The resulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regular at the endpoints. The fractional order Legendre function Pn agrees with the associated Legendre polynomial PTemplate:Su.

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • Template:Abramowitz Stegun ref2
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, Chapter 2.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Template:Dlmf
  • Template:Dlmf
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

  1. Template:Harvnb
  2. 2.0 2.1 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 Cite error: Invalid <ref> tag; name "arfken" defined multiple times with different content
  3. M. Le Gendre, "Recherches sur l'attraction des sphéroïdes homogènes," Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées, Tome X, pp. 411-435 (Paris, 1785). [Note: Legendre submitted his findings to the Academy in 1782, but they were published in 1785.] Available on-line (in French) at: http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf .
  4. Jackson, J.D. Classical Electrodynamics, 3rd edition, Wiley & Sons, 1999. page 103