Antiderivatives for Complex Functions

Since we have the same product rule, quotient rule, sum rule, chain rule etc. available to us for differentiating complex functions, we already know many antiderivatives. For example, by differentiating f (z) = z one obtains f ′(z) = nzn−1, and from this one sees that the antiderivative of z is 1 n+1 z – except for the very important case where n = −1. Of course that special case is very important in real analysis as it leads to the natural logarithm function. It will turn out to be very important in complex analysis as well, and will lead us to the complex logarithm. But this foreshadows some major differences from the real case, due to the fact that the complex exponential is not an injective function and therefore does not have an inverse function.