-----------------
#   Aufgabe 2   #
-----------------


# 2a)

u1 = function(x,y)
{
  z = x+1i*y
  res = Re(z^2)
  return(res)
}
v1 = function(x,y)
{
  z = x+1i*y
  res = Im(z^2)
  return(res)
}

# Hoehenlinien koennen mit einem Contour-Plot
# dargestellt werden: der contour()-Befehl hat 
# die Syntax 
# contour(  vector ,  vector , matrix ) = 
# contour( x-Achse , y-Achse , f(x,y) ) , 
# die f(x,y)-Werte muessen also als Matrix 
# uebergeben werden:

x = seq(from=-2,to=2,by=0.01)
y = x
nx = length(x)
ny = nx

# die Funktionswerte von u und v werden in 
# eine Matrix mat_u, mat_v geschrieben:

mat_u = matrix(0,nx,ny)
mat_v = matrix(0,nx,ny)
for(i in 1:nx)
{
  for(j in 1:ny)
  {
    mat_u[i,j] = u1(x[i],y[j])
    mat_v[i,j] = v1(x[i],y[j])
  }
}

info = "u(x,y)=const  in blau ,  v(x,y)=const  in rot"
contour( x , y , mat_u , col = "blue" , main=info )
contour( x , y , mat_v , col = "red", add=TRUE )


# 2b)

info = "u(x,y)=const  in blau ,  v(x,y)=const  in rot"
contour( x , y , mat_u , col = "blue" , nlevels=25 , main=info )
contour( x , y , mat_v , col = "red", nlevels=25 , add=TRUE )


# 2c)

# f(z) = z^3

u1 = function(x,y)
{
  z = x+1i*y
  res = Re(z^3)
  return(res)
}
v1 = function(x,y)
{
  z = x+1i*y
  res = Im(z^3)
  return(res)
}

x = seq(from=-2,to=2,by=0.01)
y = x
nx = length(x)
ny = length(y)
mat_u = matrix(0,nx,ny)
mat_v = matrix(0,nx,ny)
for(i in 1:nx)
{
  for(j in 1:ny)
  {
    mat_u[i,j] = u1(x[i],y[j])
    mat_v[i,j] = v1(x[i],y[j])
  }
}

info = "u(x,y)=const  in blau ,  v(x,y)=const  in rot"
contour( x , y , mat_u , col = "blue" , nlevels=25 , main=info )
contour( x , y , mat_v , col = "red", nlevels=25 , add=TRUE )


# f(z) = exp(z)

u1 = function(x,y)
{
  z = x+1i*y
  res = Re(exp(z))
  return(res)
}
v1 = function(x,y)
{
  z = x+1i*y
  res = Im(exp(z))
  return(res)
}

x = seq(from=-2,to=2,by=0.01)
y = x
nx = length(x)
ny = length(y)
mat_u = matrix(0,nx,ny)
mat_v = matrix(0,nx,ny)
for(i in 1:nx)
{
  for(j in 1:ny)
  {
    mat_u[i,j] = u1(x[i],y[j])
    mat_v[i,j] = v1(x[i],y[j])
  }
}

info = "u(x,y)=const  in blau ,  v(x,y)=const  in rot"
contour( x , y , mat_u , col = "blue" , nlevels=25 , main=info )
contour( x , y , mat_v , col = "red", nlevels=25 , add=TRUE )