= Solution
\b[The printed assertion is false.] Take real $w=1-\varepsilon$ with $0<\varepsilon<1$. The <principal complex logarithm> is unambiguous, but
$$
|\operatorname{Log}(1-w)|=|\log\varepsilon|\longrightarrow\infty,
\qquad |w|\longrightarrow1.
$$
No finite $C_3$ can satisfy the proposed estimate, even on these points within the logarithm's domain. At $w=1$ the logarithm is undefined, and the usual principal analytic branch also excludes its cut. These are additional reasons the literal all-$w$ assertion cannot hold.
The valid estimate for the final product proof is instead
$$
\log|1-w|\le\log(1+|w|)\le|w|\qquad(w\ne1),
$$
or equivalently $|1-w|\le e^{|w|}$ for every $w$, including $w=1$. The upper bound on the real part of the logarithm is enough for growth; a bound on the full modulus of the logarithm is unnecessary. The preceding root Solution supplies the complete convergence and exponential-growth proof using this correction.
Back to article page
