Let r>0 -π<θ<π be real. Calculate all the values of Re( (ln(z))/(z)) where z=r(cos(\theta)+sin(\theta)).

Question

Let r>0 -π<θ<π be real. Calculate all the values of


`Re( (ln(z))/(z))`

where
`z=r(cos(\theta)+sin(\theta))`.

Answer 1

For a real z, ln(z)/z is always a real number, so the Re() is quite useless.

Thus we just need to write
`ln(z)/z=ln(r)+ln(\cos(\theta)+\sin(\theta))`, and we can't simplify any more.