#--------------#
#   Blatt 10   #
#--------------#


# Aufgabe 1e)

n = 1000
theta = seq(0,pi,length=n)

Y00 = sqrt(1/4/pi)
Y10 = sqrt(3/4/pi)*( cos(theta) )
Y11 = sqrt(3/8/pi)*( sin(theta) )
Y20 = sqrt(5/16/pi)*( 3 * (cos(theta))^2 - 1 )
Y21 = sqrt(15/8/pi)*( cos(theta) * sin(theta) )
Y22 = sqrt(15/32/pi)*( sin(theta) * sin(theta) )

x00 = abs(Y00)^2 * sin(theta)
z00 = abs(Y00)^2 * cos(theta)

x10 = abs(Y10)^2 * sin(theta)
z10 = abs(Y10)^2 * cos(theta)
x11 = abs(Y11)^2 * sin(theta)
z11 = abs(Y11)^2 * cos(theta)

x20 = abs(Y20)^2 * sin(theta)
z20 = abs(Y20)^2 * cos(theta)
x21 = abs(Y21)^2 * sin(theta)
z21 = abs(Y21)^2 * cos(theta)
x22 = abs(Y22)^2 * sin(theta)
z22 = abs(Y22)^2 * cos(theta)


par(mfrow=c(2,3))

info = "polar plot |Y00|^2"
plot(x00,z00,xlim=c(-0.4,0.4),ylim=c(-0.4,0.4),main=info)
points(-x00,z00)

info = "polar plot |Y10|^2"
plot(x10,z10,xlim=c(-0.4,0.4),ylim=c(-0.4,0.4),main=info)
points(-x10,z10)

info = "polar plot |Y11|^2"
plot(x11,z11,xlim=c(-0.4,0.4),ylim=c(-0.4,0.4),main=info)
points(-x11,z11)

info = "polar plot |Y20|^2"
plot(x20,z20,xlim=c(-0.4,0.4),ylim=c(-0.4,0.4),main=info)
points(-x20,z20)

info = "polar plot |Y21|^2"
plot(x21,z21,xlim=c(-0.4,0.4),ylim=c(-0.4,0.4),main=info)
points(-x21,z21)

info = "polar plot |Y22|^2"
plot(x22,z22,xlim=c(-0.4,0.4),ylim=c(-0.4,0.4),main=info)
points(-x22,z22)



# zu Aufgabe 3)

dx = 10^(-3)
x = seq(from=-1+dx, to=1-dx, by=dx)

Q = 1/2*log( (1+x)/(1-x) )
P1 = x 
P3 = (5*x^3-3*x)/2

c1 = sum(Q*P1)*dx
c1                         # 1, nicht 0..
c3 = sum(Q*P3)*dx
c3                         # 1/6, nicht 0


require(gsl)               # hat Legendre-Polynome
?gsl
plot(x,legendre_Pl(3,x))   # ok, geht..


m = 9 
par(mfrow=c(3,3))
X = legendre_Pl(0,x)
for(k in 1:m)
{
  X = cbind( X , legendre_Pl(k,x) ) 
  res = lm( Q ~ -1+X )
  info = paste("ell =",k)
  plot(x,Q,main=info)
  points(x,res$fit,col="red",cex=0.75)
  #readline()
}