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

# 3a)

t = seq(from=0,to=1,length=1000)
phi = seq(from=pi/4,to=3*pi/4,length=1000)

z = 1-2*t+1i
z2 = sqrt(2)*exp(1i*phi)

plot(z,typ="l",col="red",xlim=c(-1.5,1.5),ylim=c(-1.5,1.5))
lines(z2,col="green")
points(1+1i,cex=1)
points(-1+1i,cex=1)
points(0+0*1i)

# Bemerkung: der Buchstabe t wird in R schon benutzt, 
# ist eigentlich schon verbraucht, der obige code 
# funktioniert trotzdem.


# 3b) 

n = 10000
t = seq( from=0, to=1, length=n )
z = 1-2*t+1i
dz = diff(z)                                        
z = z[-1]        
sum( z * dz )          # -2i bis auf discretization-error O(1/n)



# Jetzt die Wegintegrale:

# int_{gamma_red} bar(z) dz :

n = 10000
t = seq( from=0, to=1, length=n )
z = 1-2*t+1i
dz = diff(z)                                        
z = z[-1]        
sum( Conj(z) * dz )    # +2i


# int_{gamma_red} 1/z dz :

n = 10000
t = seq( from=0, to=1, length=n )
z = 1-2*t+1i
dz = diff(z)                                        
z = z[-1]        
sum( 1/z * dz )        # pi/2 *i 


# int_{gamma_green} bar(z) dz :

n = 10000
phi = seq(from=pi/4,to=3*pi/4,length=n)
z = sqrt(2)*exp(1i*phi)
dz = diff(z)                                        
z = z[-1]        
sum( Conj(z) * dz )    # pi*i


# int_{gamma_green} 1/z dz :

n = 10000
phi = seq(from=pi/4,to=3*pi/4,length=n)
z = sqrt(2)*exp(1i*phi)
dz = diff(z)                                        
z = z[-1]        
sum( 1/z * dz )        # pi/2 *i