Module displacements

Legendre polynomial.

Check equation (3) from Townsend, 2002:

>>> ls,x = [0,1,2,3,4,5],cos(linspace(0,pi,100))
>>> check = 0
>>> for l in ls:
...     for m in range(-l,l+1,1):
...         Ppos = legendre_(l,m,x)
...         Pneg = legendre_(l,-m,x)
...         mycheck = Pneg,(-1)**m * factorial(l-m)/factorial(l+m) * Ppos
...         check += sum(abs(mycheck[0]-mycheck[1])>1e-10)
>>> print check
0

Spherical harmonic according to Townsend, 2002.

This function is memoized: once a spherical harmonic is computed, the result is stored in memory

>>> theta,phi = mgrid[0:pi:20j,0:2*pi:40j]
>>> Ylm20 = sph_harm(theta,phi,2,0)
>>> Ylm21 = sph_harm(theta,phi,2,1)
>>> Ylm22 = sph_harm(theta,phi,2,2)
>>> Ylm2_2 = sph_harm(theta,phi,2,-2)
>>> p = figure()
>>> p = gcf().canvas.set_window_title('test of function <sph_harm>')
>>> p = subplot(411);p = title('l=2,m=0');p = imshow(Ylm20.real,cmap=cm.RdBu)
>>> p = subplot(412);p = title('l=2,m=1');p = imshow(Ylm21.real,cmap=cm.RdBu)
>>> p = subplot(413);p = title('l=2,m=2');p = imshow(Ylm22.real,cmap=cm.RdBu)
>>> p = subplot(414);p = title('l=2,m=-2');p = imshow(Ylm2_2.real,cmap=cm.RdBu)

Derivative of spherical harmonic wrt colatitude.

Using Y_l^m(theta,phi).

Equation:

   sin(theta)*dY/dtheta = (l*J_{l+1}^m * Y_{l+1}^m - (l+1)*J_l^m * Y_{l-1,m})

E.g.: Phd thesis of Joris De Ridder

Derivative of spherical harmonic wrt longitude.

Using Y_l^m(theta,phi).

Equation:

   dY/dphi = i*m*Y

Radial displacement, see Zima 2006.

t in phase units