Definition:Contour/Simple/Complex Plane

From ProofWiki
Jump to navigation Jump to search

Definition

Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, 2, \ldots, n}$.

Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.


$C$ is a simple contour if and only if:

$(1): \quad$ For all $i, j \in \set {1, \ldots, n}, t_1 \in \hointr {a_i} {b_i}, t_2 \in \hointr {a_j} {b_j}$ with $t_1 \ne t_2$, we have $\map {\gamma_i} {t_1} \ne \map {\gamma_j} {t_2}$.
$(2): \quad$ For all $k \in \set {1, \ldots, n}, t \in \hointr {a_k} {b_k}$ where either $k \ne 1$ or $t \ne a_1$, we have $\map {\gamma_k} t \ne \map {\gamma_n} {b_n}$.


Thus a simple contour is a contour that does not intersect itself.


Also see


Notes



Note that a simple contour may be a closed contour, so $\map {\gamma_1} {a_1} = \map {\gamma_n} {b_n}$.


Sources