Table of Contents

The Complex Natural Logarithm Function Examples 1
Recall from The Complex Natural Logarithm Function page that the complex natural logarithm function is defined for all $z \in \mathbb{C} \setminus \{ 0 \}$ by:
(1)Where $\arg (z)$ is specified in some branch $A_{y_0} = \{ z = x + yi \in \mathbb{C} : x \in \mathbb{R}, y_0 \leq y < y_0 + 2\pi \}$, i.e., $y_0 \leq \arg (z) < y_0 + 2\pi$.
We will now look at some example problems regarding the complex natural logarithm function.
Example 1
Simplify $\log (1 + i)$ for the principal branch of the logarithm function, i.e., for all $z \in \mathbb{C} \setminus \{ 0 \}$, $0 \leq \mathrm{Arg}(z) < 2\pi$.
We have that:
(2)Example 2
Find all possible values for $\log (1)$.
We have that:
(3)Note that $\arg(1) = 2k \pi$ where $k \in \mathbb{Z}$, and so for each $k \in \mathbb{Z}$:
(4)Example 3
Find all possible values for $\log (i)$.
We have that:
(5)Note that $\arg(i) = \pi + 2k\pi$ where $k \in \mathbb{Z}$, and so for each $k \in \mathbb{Z}$:
(6)