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P: 321
 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.