Principal complex logarithm

The principal complex logarithm is $\mathrm{Log}z=\log |z|+i\mathrm{Arg}z$, with the stated pointwise convention $-\pi <\mathrm{Arg}z\le \pi$. It is defined for nonzero $z$, but discontinuous across the negative real axis.

Its restriction to $\mathbb{C}\setminus (-\infty ,0]$ is a holomorphic logarithm branch with derivative $1/z$. Approaching $-1$ from above and below gives limiting imaginary parts $\pi$ and $-\pi$. Thus a pointwise principal-value convention must be distinguished from the open domain on which the selected branch is holomorphic. Products can cross the cut, so $\mathrm{Log}(zw)=\mathrm{Log}z+\mathrm{Log}w$ does not hold without a branch restriction.

The power series $\mathrm{Log}(1-w)=-{\sum }_{j\ge 1}{w}^j/j$ proves this estimate. There is no global bound $|\mathrm{Log}(1-w)|\le C|w|$, since the logarithm diverges as $w\to 1$. The different estimate $\log |1-w|\le \log (1+|w|)\le |w|$ is valid wherever $w\ne 1$, and is sufficient for growth bounds on canonical products.