# legendreP

Legendre polynomials

## Description

example

legendreP(n,x) returns the nth degree Legendre polynomial at x.

## Examples

### Find Legendre Polynomials for Numeric and Symbolic Inputs

Find the Legendre polynomial of degree 3 at 5.6.

legendreP(3,5.6)
ans =
430.6400

Find the Legendre polynomial of degree 2 at x.

syms x
legendreP(2,x)
ans =
(3*x^2)/2 - 1/2

If you do not specify a numerical value for the degree n, the legendreP function cannot find the explicit form of the polynomial and returns the function call.

syms n
legendreP(n,x)
ans =
legendreP(n, x)

### Find Legendre Polynomial with Vector and Matrix Inputs

Find the Legendre polynomials of degrees 1 and 2 by setting n = [1 2].

syms x
legendreP([1 2],x)
ans =
[ x, (3*x^2)/2 - 1/2]

legendreP acts element-wise on n to return a vector with two elements.

If multiple inputs are specified as a vector, matrix, or multidimensional array, the inputs must be the same size. Find the Legendre polynomials where input arguments n and x are matrices.

n = [2 3; 1 2];
xM = [x^2 11/7; -3.2 -x];
legendreP(n,xM)
ans =
[ (3*x^4)/2 - 1/2,        2519/343]
[           -16/5, (3*x^2)/2 - 1/2]

legendreP acts element-wise on n and x to return a matrix of the same size as n and x.

### Differentiate and Find Limits of Legendre Polynomials

Use limit to find the limit of a Legendre polynomial of degree 3 as x tends to -∞.

syms x
expr = legendreP(4,x);
limit(expr,x,-Inf)
ans =
Inf

Use diff to find the third derivative of the Legendre polynomial of degree 5.

syms n
expr = legendreP(5,x);
diff(expr,x,3)
ans =
(945*x^2)/2 - 105/2

### Find Taylor Series Expansion of Legendre Polynomial

Use taylor to find the Taylor series expansion of the Legendre polynomial of degree 2 at x = 0.

syms x
expr = legendreP(2,x);
taylor(expr,x)
ans =
(3*x^2)/2 - 1/2

### Plot Legendre Polynomials

Plot Legendre polynomials of orders 1 through 4.

syms x y
fplot(legendreP(1:4, x))
axis([-1.5 1.5 -1 1])
grid on

ylabel('P_n(x)')
title('Legendre polynomials of degrees 1 through 4')
legend('1','2','3','4','Location','best')

### Find Roots of Legendre Polynomial

Use vpasolve to find the roots of the Legendre polynomial of degree 7.

syms x
roots = vpasolve(legendreP(7,x) == 0)
roots =
-0.94910791234275852452618968404785
-0.74153118559939443986386477328079
-0.40584515137739716690660641207696
0
0.40584515137739716690660641207696
0.74153118559939443986386477328079
0.94910791234275852452618968404785

## Input Arguments

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Degree of polynomial, specified as a nonnegative number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array. All elements of nonscalar inputs should be nonnegative integers or symbols.

Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array.

collapse all

### Legendre Polynomial

• The Legendre polynomials are defined as

$P\left(n,x\right)=\frac{1}{{2}^{n}n!}\frac{{d}^{n}}{d{x}^{n}}{\left({x}^{2}-1\right)}^{n}.$

• The Legendre polynomials satisfy the recursion formula

$\begin{array}{l}P\left(n,x\right)=\frac{2n-1}{n}xP\left(n-1,x\right)-\frac{n-1}{n}P\left(n-2,x\right),\\ \text{where}\\ P\left(0,x\right)=1\\ P\left(1,x\right)=x.\end{array}$

• The Legendre polynomials are orthogonal on the interval [-1,1] with respect to the weight function w(x) = 1, where

• The relation with Gegenbauer polynomials G(n,a,x) is

$P\left(n,x\right)=G\left(n,\frac{1}{2},x\right).$

• The relation with Jacobi polynomials P(n,a,b,x) is

$P\left(n,x\right)=P\left(n,0,0,x\right).$