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legendre
associated Legendre functions
Syntax
y = legendre(n,m,x [,normflag])
Arguments
- n
non negative integer or vector of non negative integers regularly spaced with increment equal to 1
- m
non negative integer or vector of non negative integers regularly spaced with increment equal to 1
- x
real matrix(elements of
x
must be in the[-1,1]
interval)- normflag
(optional) scalar string
Description
When n
and m
are scalars,
legendre(n,m,x)
evaluates the associated Legendre
function Pnm(x) at all the elements of x
. The
definition used is :
where Pn
is the Legendre polynomial of degree
n
. So legendre(n,0,x)
evaluates the
Legendre polynomial Pn(x) at all the elements of
x
.
When the normflag is equal to "norm" you get a normalized version
(without the (-1)^m
factor), precisely :
which is useful to compute spherical harmonic functions (see Example 3):
For efficiency, one of the two first arguments may be a vector, for
instance legendre(n1:n2,0,x)
evaluates all the Legendre
polynomials of degree n1, n1+1, ..., n2 at the
elements of x
and
legendre(n,m1:m2,x)
evaluates all the Legendre
associated functions Pnm for m=m1, m1+1, ..., m2 at
x
.
Output format
In any case, the format of y
is :
and :
y(i,j) = P(n(i),m;x(j)) if n is a vector y(i,j) = P(n,m(i);x(j)) if m is a vector y(1,j) = P(n,m;x(j)) if both n and m are scalars
so that x
is preferably a row vector but any
mx x nx
matrix is expected and considered as an
1 x (mx * nx)
matrix, reshaped following the column
order.
Examples
// example 1 : plot of the 6 first Legendre polynomials on (-1,1) l = nearfloat("pred",1); x = linspace(-l,l,200)'; y = legendre(0:5, 0, x); clf() plot2d(x,y', leg="p0@p1@p2@p3@p4@p5@p6") xtitle("the 6 th first Legendre polynomials")
// example 2 : plot of the associated Legendre functions of degree 5 l = nearfloat("pred",1); x = linspace(-l,l,200)'; y = legendre(5, 0:5, x, "norm"); clf() plot2d(x,y', leg="p5,0@p5,1@p5,2@p5,3@p5,4@p5,5") xtitle("the (normalized) associated Legendre functions of degree 5")
// example 3 : define then plot a spherical harmonic // 3-1 : define the function Ylm function [y]=Y(l, m, theta, phi) // theta may be a scalar or a row vector // phi may be a scalar or a column vector if m >= 0 then y = (-1)^m/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, m, cos(theta), "norm") else y = 1/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, -m, cos(theta), "norm") end endfunction // 3.2 : define another useful function function [x, y, z]=sph2cart(theta, phi, r) // theta row vector 1 x nt // phi column vector np x 1 // r scalar or np x nt matrix (r(i,j) the length at phi(i) theta(j)) x = r.*(cos(phi)*sin(theta)); y = r.*(sin(phi)*sin(theta)); z = r.*(ones(phi)*cos(theta)); endfunction // 3-3 plot Y31(theta,phi) l = 3; m = 1; theta = linspace(0.1,%pi-0.1,60); phi = linspace(0,2*%pi,120)'; f = Y(l,m,theta,phi); [x1,y1,z1] = sph2cart(theta,phi,abs(f)); [xf1,yf1,zf1] = nf3d(x1,y1,z1); [x2,y2,z2] = sph2cart(theta,phi,abs(real(f))); [xf2,yf2,zf2] = nf3d(x2,y2,z2); [x3,y3,z3] = sph2cart(theta,phi,abs(imag(f))); [xf3,yf3,zf3] = nf3d(x3,y3,z3); clf() subplot(1,3,1) plot3d(xf1,yf1,zf1,flag=[2 4 4]); xtitle("|Y31(theta,phi)|") subplot(1,3,2) plot3d(xf2,yf2,zf2,flag=[2 4 4]); xtitle("|Real(Y31(theta,phi))|") subplot(1,3,3) plot3d(xf3,yf3,zf3,flag=[2 4 4]); xtitle("|Imag(Y31(theta,phi))|")
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