We know that d/dz (zlog(z) - z) = log(z).
If this is true then the integral of log (z) is zlog(z) - z.
You have the points so:
[z1log (z1) - z1] - [z2log (z2) - z2]
where z1= 1 + i and z2= 1 - i
Evaluate the line integral log(z) dz
where C is the arc defined by z= root2 x e^(itheta), -pi/4 < or = theta < or = pi/4.
between the points (1-i) and (1+i).
In a domain where log(z) is analytic we know that d/dz (zlog(z) - z) = log(z).
Hence evaluate the above integral directly.