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

 $\ds \cmod {1 - \frac {\map \ln {1 + z} } z}$ $=$ $\ds \cmod {\sum_{n \mathop = 2}^\infty \frac {\paren {-1}^n z^{n - 1} }n}$ Linear Combination of Convergent Series $\ds$ $\le$ $\ds \sum_{n \mathop = 2}^\infty \cmod {\frac {\paren {-z}^{n - 1} } n}$ Triangle Inequality for Series $\ds$ $\le$ $\ds \sum_{n \mathop = 2}^\infty \frac {\cmod z^{n - 1} } 2$ $n \ge 2$ $\ds$ $=$ $\ds \frac {\cmod z / 2} {1 - \cmod z}$ Sum of Infinite Geometric Sequence $\ds$ $\le$ $\ds \frac 1 2$ $\cmod z \le \dfrac 1 2$

By the Triangle Inequality:

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

$\blacksquare$