legendreP

Find the Legendre polynomial of degree 3 at 5.6.

y = legendreP(3,5.6)

y = 
430.6400

Find the Legendre polynomial of degree 2 at x.

syms x
y = legendreP(2,x)

y = 
$$\frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}$$

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
y = legendreP(n,x)

y = $$\mathrm{legendreP}\left(n,x\right)$$

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

syms x
y = legendreP([1 2],x)

y = 
$$\left(\begin{array}{cc}x& \frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}\end{array}\right)$$

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];
yM = legendreP(n,xM)

yM = 
$$\left(\begin{array}{cc}\frac{3\hspace{0.17em}{x}^4}{2}-\frac{1}{2}& \frac{2519}{343}\\ -\frac{16}{5}& \frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}\end{array}\right)$$

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

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

syms x
P3 = legendreP(3,x);
P3 = limit(P3,x,-Inf)

P3 = $$-\infty$$

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

P5 = legendreP(5,x);
D3P5 = diff(P5,x,3)

D3P5 = 
$$\frac{945\hspace{0.17em}{x}^2}{2}-\frac{105}{2}$$

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

syms x
y = legendreP(2,x)

y = 
$$\frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}$$

t = taylor(y,x)

t = 
$$\frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}$$

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')

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

syms x
roots = vpasolve(legendreP(7,x) == 0)

roots = 
$$\left(\begin{array}{c}-0.94910791234275852452618968404785\\ -0.74153118559939443986386477328079\\ -0.40584515137739716690660641207696\\ 0\\ 0.40584515137739716690660641207696\\ 0.74153118559939443986386477328079\\ 0.94910791234275852452618968404785\end{array}\right)$$