Quote by latentcorpse
i see that if we take the principal argument of z and hence the proncipal value of log it does indeed jump as you cross the negative real axis  why does this mean it can't be extended to a holomorphic function on C\0?

Holomorphic means in particular continuous. No matter how the values on the negative real axis are defined, there will always be a discontinuity. See also the discussion on the
complex logarithm.
also did i set up the cauchy riemann bit ok in my first post?

Looks correct.