Complex Analysis/Path

Definition: Path

Let $U\subset \mathbb {C}$ be a subset. A path in $U$ is a continuous mapping with:

$\gamma \colon [a,b]\rightarrow U$ with $a<b$ and $a,b\in \mathbb {R}$.

Definition: Trace of a Path

The trace of a path $\gamma \colon [a,b]\rightarrow U$ in $U\subset \mathbb {C}$ is the image of the function $\gamma$:

$\mathrm {Trace} (\gamma ):={\gamma (t)\in \mathbb {C} \ |\ t\in [a,b]}$

Definition: Closed Path

Let $\gamma \colon [a,b]\rightarrow U$ be a path in $U\subset \mathbb {C}$. The mapping $\gamma$ is called a closed path if:

$\gamma (a)=\gamma (b)$

Definition: Region

Let $U\subset \mathbb {C}$ be an open subset of $\mathbb {C}$. Then $U$ is called a region.

Definition: Path-Connected

Let $U\subset \mathbb {C}$ be a non-empty set.

$U$ is path-connected $:\Longleftrightarrow \ \forall _{z_{1},z_{2}\in U}\exists _{\gamma \colon [a,b]\rightarrow U}:\ \gamma (a)=z_{1}\wedge \gamma (b)=z_{2}\wedge Trace(\gamma )\subseteq U$

Definition: Domain

Let $G\subset \mathbb {C}$ be a non-empty subset of $\mathbb {C}$. If

$G$ is open

$G$ is path-connected

Then $G$ is called a domain in $\mathbb {C}$.

Example (Circular Paths)

Let $z_{o}\in \mathbb {C}$ be a complex number, and let $r>0$ be a radius. A circular path $\gamma _{z_{o},r}\colon [0,2\pi ]\rightarrow \mathbb {C}$ around $z_{o}\in \mathbb {C}$ is defined as:

$\gamma _{z_{o},r}(t):=z_{o}+r\cdot e^{i\cdot t}$

Example - Paths with Ellipse as Trace

Let $z_{o}\in \mathbb {C}$ be a complex number, and let $a,b>0$ be the semi-axes of an ellipse. An elliptical path $\gamma _{z_{o},a,b}\colon [0,2\pi ]\rightarrow \mathbb {C}$ around $z_{o}\in \mathbb {C}$ is defined as:

$\gamma _{z_{o},a,b}(t):=z_{o}+a\cdot \cos(t)+i\cdot b\cdot \sin(t)$

Gardener's Construction of an Ellipse

Convex Combinations

Let $z_{1},z_{2}\in \mathbb {C}$ be complex numbers, and let $t\in [0,1]$ be a scalar. A path $\gamma _{z_{1},z_{2}}\colon [0,1]\rightarrow \mathbb {C}$ is defined such that its trace is the line segment connecting $z_{1},z_{2}\in \mathbb {C}$:

$\gamma _{z_{1},z2}(t):=(1-t)\cdot z_{1}+t\cdot z_{2}$

Such a path is called a convex combination of the first order (see also Higher-Order Convex Combinations).

Animation of a Convex Combination of Two Vectors as Mapping

Convex Combination as Mapping in an Animated GIF

Integration Path

Let $G\subset \mathbb {C}$ be a domain. An integration path in $G$ is a path that is piecewise continuously differentiable with

$\gamma \colon [a,b]\rightarrow U$ with $a<b$ and $a,b\in \mathbb {R}$.

Remark

An integration path can, for example, be expressed piecewise as convex combinations between multiple points $z_{1},\ldots z_{n}\in \mathbb {C}$. The overall path does not need to be differentiable at points $z_{1},\ldots z_{n}\in \mathbb {C}$. The trace of such a path is also called a polygonal path.

See Also

Ellipse

Convex Combination

Paths in Topological Vector Spaces

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