Bounds for Complex Logarithm

Theorem
Let $\ln$ denote the complex logarithm.

Let $z \in \C$ with $\cmod z \le \dfrac 1 2$.

Then:
 * $\dfrac 1 2 \cmod z \le \cmod {\map \ln {1 + z} } \le \dfrac 3 2 \cmod z$

Proof
By definition of complex logarithm:
 * $-\map \ln {1 + z} = \ds \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} n$

Thus

By the Triangle Inequality:
 * $\dfrac 1 2 \le \cmod {\dfrac {\map \ln {1 + z} } z} \le \dfrac 3 2$

Also see

 * Bounds of Natural Logarithm
 * Bounds for Complex Exponential