---------------
#  Aufgabe 1  #
---------------

# 1a)
Re( (1i)^275 )
Im( (1i)^275 )

# 1b)
Re( 1/(1i)^5 )
Im( 1/(1i)^5 )

# 1c)
Re( (1i-1)^3 )
Im( (1i-1)^3 )

# 1d)
Re( (1+2*1i)/(3-4*1i) )
Im( (1+2*1i)/(3-4*1i) )

# 1e)
Re( 1/(1+1i)^2 )
Im( 1/(1+1i)^2 )


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

# 2a)
Arg(1-1i)
Arg(1-1i)/pi

# 2b)
Arg(-sqrt(3)+1i)
Arg(-sqrt(3)+1i)/pi

# 2c)
Arg( (1-1i)^3 )
Arg( (1-1i)^3 )/pi

# 2d)
Arg( 2/(1+sqrt(3)*1i) )
Arg( 2/(1+sqrt(3)*1i) )/pi

# 2e)
Arg( 2/(1i-1) )
Arg( 2/(1i-1) )/pi


---------------
#  Aufgabe 3  #
---------------

z = (4+2*1i)/5
k = 0:100
powers = z^k
sn = cumsum(powers)
plot(sn)

z0 = 1/(1-z)
points(z0,col="red",pch="X")



---------------
#  Aufgabe 4  #
---------------

# 4a)
dx = 0.01
dy = 0.01
x = seq(from=0,to=0.5,by=dx)
y = seq(from=0,to=2.0,by=dy)

z = 1i*y
plot(z,type="l",xlim=c(-1,1),ylim=c(0,2))
for(k in 1:5)
{
  lines(z+k*0.1)
}
for(k in 0:20)
{
  lines(x+1i*k*0.1)
}

# 4b)
dx = 0.01
dy = 0.01
x = seq(from=0,to=0.5,by=dx)
y = seq(from=0,to=2.0,by=dy)

z = 1i*y
plot(z^2,type="l",xlim=c(-1,1),ylim=c(0,2))
for(k in 1:5)
{
  lines( (z+k*0.1)^2 )
}
for(k in 0:20)
{
  lines( (x+1i*k*0.1)^2 )
}

# wenn wir die x-Achse von -4 bis +1 gehen lassen wollen:
z = 1i*y
plot(z^2,type="l",xlim=c(-4,1),ylim=c(-1,4))
for(k in 1:5)
{
  lines( (z+k*0.1)^2 )
}
for(k in 0:20)
{
  lines( (x+1i*k*0.1)^2 )
}