Legendre polynomial

The Legendre polynomials (after Adrien-Marie Legendre ), also called zonal spherical functions , are special polynomials that form an orthogonal system of functions on the interval . They are the particular solutions of the legendary differential equation . The Legendre polynomials play an important role in theoretical physics , especially in electrodynamics and quantum mechanics , as well as in the field of filter technology for Legendre filters . ${\displaystyle [-1,1]}$

Differential equation and polynomials

Legendre's differential equation

The legendary differential equation

${\displaystyle \left(1-x^{2}\right)\,y''(x)-2x\,y'(x)+n(n+1)\,y(x)=0}$

can also be used as an ordinary linear differential equation of the second order in the form

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[\left(1-x^{2}\right)\,y'(x)\right]+n(n+1)\,y(x)=0}$

for and are represented. ${\displaystyle x\in [-1,1]}$ ${\displaystyle n\in \mathbb {N} _{0}}$

It is a special case of the Sturm-Liouville differential equation

${\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}\left((1-x^{2}){\frac {\mathrm {d} y}{\mathrm {d} x}}\right)=n(n+1)y.}$

The general solution to this differential equation is

${\displaystyle y(x)=A\,P_{n}(x)+B\,Q_{n}(x)}$

with the two linearly independent functions and . The Legendre polynomials are therefore also referred to as Legendre functions of the first type and Legendre functions of the second type, because these are no longer polynomials. ${\displaystyle P_{n}(x)}$ ${\displaystyle Q_{n}(x)}$ ${\displaystyle P_{n}(x)}$ ${\displaystyle Q_{n}(x)}$

In addition, there is a generalized Legendre differential equation, the solutions of which are called Legendre polynomials .

First polynomials

The first six Legendre polynomials

The first Legendre polynomials are:

${\displaystyle P_{0}(x)=1\,}$

${\displaystyle P_{1}(x)=x\,}$

${\displaystyle P_{2}(x)={\frac {1}{2}}(3x^{2}-1)}$

${\displaystyle P_{3}(x)={\frac {1}{2}}(5x^{3}-3x)}$

${\displaystyle P_{4}(x)={\frac {1}{8}}(35x^{4}-30x^{2}+3)}$

${\displaystyle P_{5}(x)={\frac {1}{8}}(63x^{5}-70x^{3}+15x)}$

${\displaystyle P_{6}(x)={\frac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)}$

The -th Legendre polynomial is ${\displaystyle n}$

${\displaystyle P_{n}(x)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {(2n-2k)!\ }{(n-k)!\ (n-2k)!\ k!\ 2^{n}}}x^{n-2k}}$

with the Gaussian bracket

${\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor ={\begin{cases}{\frac {n}{2}}&n\ {\text{gerade}}\\{\frac {n-1}{2}}&n\ {\text{ungerade}}\end{cases}}}$

The -th Legendre polynomial has degree and is off , i.e. i.e., it has rational coefficients. There are several forms of representation for the Legendre polynomials. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} [x]}$

Construction of orthogonal polynomials

For an interval and a weight function given on it , a sequence of real polynomials is orthogonal if they meet the orthogonality condition ${\displaystyle I=[a,b]}$ ${\displaystyle w(x)}$ ${\displaystyle (P_{n})}$ ${\displaystyle P_{n}\in \mathbb {R} [X]}$

${\displaystyle \int \limits _{a}^{b}w(x)\,P_{n}(x)\,P_{m}(x)\,\mathrm {d} x=0}$

for everyone with fulfilled. ${\displaystyle m,n\in \mathbb {N} _{0}}$ ${\displaystyle m\neq n}$

For the interval together with the simplest of all weight functions , such orthogonal polynomials can be iteratively generated with the aid of the Gram-Schmidt orthogonalization method starting from the monomials . The Legendre polynomials result if this is additionally required. ${\displaystyle I=[-1,1]}$ ${\displaystyle w(x)=1}$ ${\displaystyle (x^{n})_{n\in \mathbb {N} }}$ ${\displaystyle P_{n}(1)=1}$

properties

Rodrigues formula

${\displaystyle P_{n}(x)={\frac {1}{2^{n}\,n!}}\cdot {\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}{\bigg (}(x^{2}-1)^{n}{\bigg )}}$

The Rodrigues formula can be evaluated with the formula of Faà di Bruno and again receives the explicit form of the -th Legendre polynomial. ${\displaystyle n}$

Integral representation

For all true ${\displaystyle x\in \mathbb {C} \setminus \{+1,-1\}}$

${\displaystyle P_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \varphi \right)^{n}\,\mathrm {d} \varphi }$

Recursion formulas

The following recursion formulas apply to the Legendre polynomials:

${\displaystyle {\begin{aligned}(n+1)P_{n+1}(x)&=(2n+1)xP_{n}(x)-nP_{n-1}(x)\,\,\,\,\,\quad \quad (n=1,2,\ldots ;P_{0}=1;P_{1}=x)\\(x^{2}-1){\frac {\mathrm {d} }{\mathrm {d} x}}P_{n}(x)&=nxP_{n}(x)-nP_{n-1}(x)\end{aligned}}}$

The first recursive formula can be represented by means of the substitution in the following, frequently found way: ${\displaystyle n'=n+1}$

${\displaystyle nP_{n}(x)=(2n-1)xP_{n-1}(x)-(n-1)P_{n-2}(x)\,\,\,\,\,\qquad (n=2,3,\ldots ;P_{0}=1;P_{1}=x)}$

By using the derivation rule for expressions of the type with or , the following recursive representation of the Legendre polynomials results, which also takes the derivatives of these polynomials into account: ${\displaystyle y=x^{n}}$ ${\displaystyle y'=nx^{n-1}=nx^{-1}y}$ ${\displaystyle y^{(m)}=(n-m+1)x^{-1}y^{(m-1)}}$

${\displaystyle (n-m)P_{n}^{(m)}(x)=(2n-1)xP_{n-1}^{(m)}(x)-(n-1+m)P_{n-2}^{(m)}(x)\,\,\,\,\,\qquad (n>1;\,\,m=0\ldots n-1)}$

The initial conditions are and . ${\displaystyle P_{m}^{(m)}(x)={\frac {(2m)!}{2^{m}m!}}}$ ${\displaystyle P_{k}^{(m)}(x)={0}\,\,\,\,\,\qquad (k<m)}$

In turn, the formula given above results with its initial conditions. ${\displaystyle m=0}$

Complete orthogonal system

Consider the Hilbert space of the square integrable on defined real-valued functions equipped with the scalar product ${\displaystyle V:=L^{2}([-1,1];\mathbb {R} )}$ ${\displaystyle [-1,1]}$

${\displaystyle \langle f,g\rangle =\int _{-1}^{1}f(x)g(x)\mathrm {d} x}$.

The family of Legendre polynomials builds on a complete orthogonal system, so they are a special case of orthogonal polynomials . If these are normalized, they form a complete orthonormal system . ${\displaystyle (P_{n})_{n}}$ ${\displaystyle (V,\langle \cdot ,\cdot \rangle )}$ ${\displaystyle V}$

It applies

${\displaystyle \int \limits _{-1}^{1}P_{n}(x)P_{m}(x)\,\mathrm {d} x={\frac {2}{2n+1}}\delta _{nm}}$,

where the Kronecker delta denotes. Completeness means that every function can be "developed" in the standard topology generated by Legendre polynomials: ${\displaystyle \delta _{nm}}$ ${\displaystyle f\in V}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

${\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}\,P_{n}(x)}$

with the expansion coefficients

${\displaystyle c_{n}={\frac {2\,n+1}{2}}\,\int \limits _{-1}^{1}f(x)\,P_{n}(x)\,\mathrm {d} x.}$

In the physical or technical literature, completeness is often written as a distribution equation as follows :

${\displaystyle \sum _{n=0}^{\infty }{\frac {2\,n+1}{2}}\,P_{n}(x')\,P_{n}(x)=\delta (x'-x)}$,

where is the diracian delta distribution . Such a distribution equation should always be read in such a way that both sides of this equation can be applied to test functions. If you apply the right side to such a test function , you get . To use the left side, you have to multiply by by definition and then integrate via . But then you get exactly the above development formula (with instead of ). Orthogonality and completeness can therefore be written briefly and concisely as follows: ${\displaystyle \delta }$ ${\displaystyle x\mapsto f(x)}$ ${\displaystyle f(x')}$ ${\displaystyle f(x)}$ ${\displaystyle x}$ ${\displaystyle x'}$ ${\displaystyle x}$

• Orthogonality: for . ${\displaystyle \langle P_{n},P_{m}\rangle =0}$ ${\displaystyle m\neq n}$
• Completeness: for everyone (in terms of convergence). ${\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}\langle f,P_{n}\rangle \,P_{n}(x)}$ ${\displaystyle f\in L^{2}([-1,1];\mathbb {R} )}$ ${\displaystyle L^{2}}$

zeropoint

${\displaystyle P_{n}(x)}$has exactly simple zeros on the interval . They are symmetrical about the origin of the abscissa, since Legendre polynomials are either even or odd. There is exactly one zero of between two neighboring zeros of . The ratio in which a zero of divides the interval between two zeros of , or vice versa up to the outer of , is very variable. ${\displaystyle I=[-1,1]}$ ${\displaystyle n}$ ${\displaystyle P_{n}(x)}$ ${\displaystyle P_{n-1}(x)}$ ${\displaystyle P_{n-1}(x)}$ ${\displaystyle P_{n}(x)}$ ${\displaystyle P_{n}(x)}$

The determination of the zeros of the Legendre polynomials is a frequent task in numerical mathematics , as they play a central role in the Gauss-Legendre quadrature or the development of “any” functions according to polynomials mentioned under “Complete Orthogonal System” . There are numerous tables for this, but their use is often associated with inconvenience, because a large number of tables with suitable accuracy would have to be available for a flexible reaction. When searching for zeros, knowledge of the interval is only of limited value when choosing an iteration start, especially since knowledge of the zeros of another polynomial is also required. An approximation of the -th zero of , which becomes more and more precise, is given by: ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle x_{k}}$ ${\displaystyle P_{n}(x)}$

${\displaystyle x_{k}\approx \cos \left(\pi \,{\frac {4k-1}{4n+2}}\right),\quad k=1,\ldots ,n.}$

For example , all zeros are estimated with an accuracy of at least two decimal places, with errors between and , while the smallest zero interval is only . At three decimal places are already safe, with errors between and , while the best nesting is through only . The maximum estimation error for is only with the two fifth zeros from outside, the exact amount of which begins with . ${\displaystyle P_{10}(x)}$ ${\displaystyle 0{,}00102}$ ${\displaystyle 0{,}00016}$ ${\displaystyle P_{9}(x)}$ ${\displaystyle 0{,}13}$ ${\displaystyle P_{20}(x)}$ ${\displaystyle 0{,}00028}$ ${\displaystyle 0{,}00002}$ ${\displaystyle P_{19}(x)}$ ${\displaystyle 0{,}032}$ ${\displaystyle P_{200}(x)}$ ${\displaystyle 0{,}0000031}$ ${\displaystyle 0{,}99722851428\ldots }$

With such a start value and the first two “recursion formulas” , both the function value and its derivation can be determined with one calculation. With the help of the Newton method , all zeros except the two outer ones can be found with more than quadratic convergence, since the zeros are in the immediate vicinity of the turning points. The two outer zeros converge "only" quadratically, i. H. an initial distance from the zero point decreases after one iteration to approximately , then to and . ${\displaystyle 0{,}00102}$ ${\displaystyle 0{,}00102^{2}}$ ${\displaystyle 0{,}00102^{4},0{,}00102^{8}}$ ${\displaystyle 0{,}00102^{16}}$

The given estimate is part of a very short algorithm that supplies both all zeros of a Legendre polynomial and the appropriate weights for the Gauss-Legendre quadrature.

General properties

The following applies to each and every : ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x\in [-1,1]}$

${\displaystyle {\begin{aligned}&P_{n}(1)=1\\&P_{n}(-x)=(-1)^{n}\,P_{n}(x)\\&P_{2n}(0)=(-1)^{n}{\frac {1\cdot 3\cdot \ldots \cdot (2n-1)}{2\cdot 4\cdot \ldots \cdot 2n}}\\&P_{2n+1}(0)=0\end{aligned}}}$

Generating function

For all , , applies ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle |z|<1}$

${\displaystyle (1-2xz+z^{2})^{-1/2}=\sum _{n=0}^{\infty }P_{n}(x)z^{n}\ .}$

The power series on the right-hand side has 1 for the radius of convergence . ${\displaystyle -1\leq x\leq +1}$

The function is therefore called the generating function of the Legendre polynomials . ${\displaystyle z\mapsto (1-2xz+z^{2})^{-1/2}}$ ${\displaystyle P_{n}}$

The term that often occurs in physics (e.g. in the potentials of Newtonian gravity or electrostatics ; multipole expansion) can thus be expanded into a power series for : ${\displaystyle 1/|{\vec {x}}-{\vec {x}}\,'|}$ ${\displaystyle {\tfrac {|{\vec {x}}\,'|}{|{\vec {x}}|}}={\tfrac {r\,'}{r}}<1}$

${\displaystyle {\begin{aligned}{\frac {1}{|{\vec {x}}-{\vec {x}}\,'|}}&={\frac {1}{\sqrt {{\vec {x}}\,^{2}-2{\vec {x}}\cdot {\vec {x}}\,'+{\vec {x}}\,'^{2}}}}={\frac {1}{\sqrt {r^{2}-2rr\,'\cos \alpha +r\,'^{2}}}}={\frac {1}{r{\sqrt {1-2{\frac {r'}{r}}\cos \alpha +({\frac {r'}{r}})^{2}}}}}\\&={\frac {1}{r}}\sum _{n=0}^{\infty }\left({\frac {r\,'}{r}}\right)^{n}P_{n}(\cos \alpha )\end{aligned}}}$

Legendre functions of the 2nd kind

The first five Legendre functions of the 2nd kind

The recursion formulas of the Legendre polynomials also apply to the Legendre functions of the 2nd kind, so that they can be determined iteratively by specifying the first:

${\displaystyle Q_{0}(x)={\frac {1}{2}}\,\ln \left({\frac {1+x}{1-x}}\right)=\operatorname {artanh} (x)}$

${\displaystyle Q_{1}(x)={\frac {x}{2}}\,\ln \left({\frac {1+x}{1-x}}\right)-1=x\operatorname {artanh} (x)-1}$

${\displaystyle Q_{2}(x)={\frac {3\,x^{2}-1}{4}}\,\ln \left({\frac {1+x}{1-x}}\right)-{\frac {3\,x}{2}}={\frac {3}{2}}\left(\left(x^{2}-{\frac {1}{3}}\right)\operatorname {artanh} (x)-x\right)}$

${\displaystyle Q_{3}(x)={\frac {5\,x^{3}-3\,x}{4}}\,\ln \left({\frac {1+x}{1-x}}\right)-{\frac {5\,x^{2}}{2}}+{\frac {2}{3}}}$

The main branch is to be used for the logarithm , which results in singularities at and in the complex plane branch sections along and . ${\displaystyle x=\pm 1}$ ${\displaystyle (-\infty ,-1)}$ ${\displaystyle (1,\infty )}$

application areas

Among other things, the Legendre polynomial is used for simulations of spherical spheres, for example to determine the Taylor angle in the Taylor cone , which is the basis for electrospinning the geometry.

Web links

• Eric W. Weisstein : Legendre Polynomial . In: MathWorld (English).
• JB Calvert: Legendre Polynomials . (English)

Individual evidence

1. ↑ Numerical Recipes : Code excerpt from Numerical Recipes in C, page 152 : "z = cos (3.141592654 * (i-0.25) / (n + 0.5));"
2. ↑ Abramowitz-Stegun : Handbook of Mathematical Functions . Asymptotic development of the zeros in formula 22.16.6, page 787
3. ↑ Branch Cut . Wolfram Research, accessed September 19, 2018.