A theorem in my textbook states the following: For every n=0,±1, ±2, --- the formula ln z=Ln z ± 2nπi defines a function, which is analytic, except at 0 and on the negative real axis, and has the derivative (ln z)'=1/z. I don't understand why the logarithm isn't analytic for negative real values. It is after all defined for negative values, contrary to the real logarithm. Is it because it is not differentiable for negative real values? And if so, why?