Logarithm

For a non-zero complex number $z$, $\log z$ is the set of all $w \in \mathbb{C}$ with $e^w = z$.

Writing $z = re^{i\theta}$ with $r = |z|$ and $w = u + iv$, the condition $e^w = z$ gives $e^u = r$ and $v \in \arg z$. So

Here:

$\operatorname{Arg} z$

principal argument of $z$, the unique argument with $-\pi < \operatorname{Arg} z \le \pi$

$\arg z$

$\{\operatorname{Arg} z + 2k\pi : k \in \mathbb{Z}\}$

$\log z$ is multi-valued.

Principal Value

The logarithm value deduced by setting $k=0$.

Below you can change $z$ and see how $\operatorname{Log}(z)$ changes.

Analyticity

$\operatorname{Log} z$ is analytic on $\mathbb{C} \setminus (-\infty, 0]$ and

$\operatorname{Log} z$ is not analytic at any point of $(-\infty, 0]$.

• At $z = 0$ it is undefined.
• On the negative real axis $\operatorname{Arg} z$ jumps by $2\pi$, so $\operatorname{Log} z$ is not continuous there.

The negative real axis is the branch cut of $\operatorname{Log} z$. Any single-valued branch of $\log z$ is analytic on the plane minus a ray from $0$.

$0$ is a branch point. No branch is analytic on a neighbourhood of $0$ with only $0$ removed, since going once around $0$ changes $\arg z$ by $2\pi$.

Properties

Let $z_1$ and $z_2$ be non-zero.

• $\log(z_1 z_2) = \log z_1 + \log z_2$
• $\log\dfrac{z_1}{z_2} = \log z_1 - \log z_2$
• $\log\dfrac{1}{z} = -\log z$

Examples

$\log(1+i)$

$\log i$