Principal branch

Definition

Let $F$ be a multivalued function on a set $X \subseteq \mathbb{C}$, so that $F(z)$ is a set of complex numbers for each $z \in X$. A principal branch of $F$ is a function $f$ on $X$ with $f(z) \in F(z)$ for each $z \in X$, chosen by a standard convention. Its value $f(z)$ is the principal value of $F$ at $z$. For the main functions of complex analysis, the conventions are these. Let $z \neq 0$, and let $\operatorname{Arg} z \in (-\pi, \pi]$ be the principal value of the argument.

• The principal logarithm is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$, where $\ln$ is the real natural logarithm.
• The principal square root is $\sqrt{z} = e^{\frac{1}{2} \operatorname{Log} z}$.
• For $w \in \mathbb{C}$, the principal power is $z^w = e^{w \operatorname{Log} z}$.

Remarks

The imaginary part of $\operatorname{Log} z$ is in $(-\pi, \pi]$. For example, $\operatorname{Log} i = i\pi/2$ and $\operatorname{Log}(-1) = i\pi$. These principal branches are continuous and holomorphic on $\mathbb{C} \setminus (-\infty, 0]$, see the principal logarithm is holomorphic off the branch cut. They are not continuous at the points of $(-\infty, 0)$. Some authors use $\operatorname{Arg} z \in [0, 2\pi)$ instead. This gives a different principal branch.

Explanations

L1 human

A multivalued function gives many values at a point. The principal branch is the rule that everybody agrees on for picking one value. For the logarithm, the rule is to take the angle between $-180$ and $180$ degrees.