UEC Education System

球面調和関数 Y_lm(θ,φ)

#
# Spherical Harmonics Y_lm (t,p) t=\theta p=\varphi 
#
# Usage Y(l,m,t,p);     By R. Saito 95-6-28
#
#
#
# P(n,x) nth Legendre polynomial.
#
with(orthopoly):
#
# Pa(n,m,x) original associated Legendre polynomial.
#
pa:=proc(n,m,x)
local a,z:
if m=0 then P(n,x):
else
a:=(1 - z^2 )^(m/2) * diff(P(n,z),z$m):
z:=x:
eval(a):
fi:
end:
#
# Pa(n,m,x) associated Legendre polynomial only for Ylm.
#
pa1:=proc(n,m,x)
local a,z:
if m=0 then P(n,cos(x)):
else
a:= sin(x)^m * diff(P(n,z),z$m):
z:=cos(x):
#
# Note: eval is necessary in the folloing expression.
#
eval(a):
fi:
end:
#
# Ylm
#
Y:=proc(l,m,t,p)
sqrt( (2*l+1)/(4*Pi) * (l-abs(m))!/(l+abs(m))! )
* pa1(l,abs(m),t) * exp(I*m*p):
end: