Continuous logarithms and continuous arguments along a contour

Let $\gamma :[a,b]\to \mathbb{C}$ be a complex contour (Rectifiable complex contours, reversal, concatenation, closedness, and orientation) with trace ${\gamma }^{\ast }$, and let $p\in \mathbb{C}$ with $p\mathrm{\notin }{\gamma }^{\ast }$.

A continuous logarithm of $\gamma -p$ along $\gamma$ is a continuous function $\lambda :[a,b]\to \mathbb{C}$ with

$\exp (\lambda (t))=\gamma (t)-p\text{for every }t\in [a,b],$

the exponential being that of The complex exponential by its power series. The associated continuous argument of $\gamma -p$ along $\gamma$ is $\theta :=\mathrm{Im}\lambda :[a,b]\to \mathbb{R}$ (Real and imaginary parts, complex conjugation, and modulus).

Let $V\subseteq \mathbb{C}$ be open with $p\mathrm{\notin }V$. A holomorphic logarithm branch of $z-p$ on $V$ is a holomorphic function $L:V\to \mathbb{C}$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions) with

$\exp (L(z))=z-p\text{for every }z\in V\mathrm{.}$

These are two different objects and only the first is unconditional. A continuous logarithm along $\gamma$ is a function of the parameter $t$; it exists for every complex contour missing $p$, and prescribing the single value $\lambda (a)$ among the complex numbers whose exponential is $\gamma (a)-p$ determines it, both by Every contour missing a point admits a continuous logarithm, unique up to a constant in $2\pi i\mathbb{Z}$ ↗. A holomorphic logarithm branch is a function on a plane set, and for a general open $V$ missing $p$ there need be none.

Along a contour, whose parameter interval is connected, two continuous logarithms differ by one additive constant in $2\pi i\mathbb{Z}$. On a general open set $V$, two holomorphic branches differ by a locally constant $2\pi i\mathbb{Z}$-valued function, hence by one such constant on each connected component; a single global constant is forced only when $V$ is connected. This follows from $\ker (\exp )=2\pi i\mathbb{Z}$ ($\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$). For $p=0$ and $V$ the slit plane, Complex logarithms, the principal logarithm, and principal and multivalued complex powers names the principal logarithm; its holomorphy on that domain is proved later on this page.

A continuous argument $\theta$ carries no normalisation of its own: adding $2\pi k$ to $\theta$ for a fixed integer $k$ replaces $\lambda$ by $\lambda +2\pi ik$, which is again a continuous logarithm. What is unambiguous is the increment $\theta (b)-\theta (a)$, since the two choices differ by the same constant at both endpoints.