Slit plane: Difference between revisions
(New page: ==Definition== The '''slit plane''' is defined as the following open subset of <math>\mathbb{C}</math>: <math>\mathbb{C} \setminus \{ z \in \R \mid z \le 0 \}</math> In other words, it ...) |
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In other words, it is the complement in <math>\mathbb{C}</math> of the half-line of nonpositive reals. | In other words, it is the complement in <math>\mathbb{C}</math> of the half-line of nonpositive reals. | ||
The slit plane is a [[star-like domain]], with 1 as a star point. In particular, it is simply connected, and admits a holomorphic logarithm, given by: | The slit plane is a [[star-like domain]], with 1 as a star point. In particular, it is simply connected, and admits a [[holomorphic logarithm of an open subset|holomorphic logarithm]], given by: | ||
<math>re^{i\theta} \mapsto \log r + i\theta</math> | <math>re^{i\theta} \mapsto \log r + i\theta</math> | ||
where <math>\theta \in (-\pi,\pi)</math> is the principal argument. | where <math>\theta \in (-\pi,\pi)</math> is the principal argument. | ||
The slit plane also admits a [[holomorphic squareroot]] and holomorphic <math>n^{th}</math> roots for higher <math>n</math>. | |||
Revision as of 22:28, 26 April 2008
Definition
The slit plane is defined as the following open subset of :
In other words, it is the complement in of the half-line of nonpositive reals.
The slit plane is a star-like domain, with 1 as a star point. In particular, it is simply connected, and admits a holomorphic logarithm, given by:
where is the principal argument.
The slit plane also admits a holomorphic squareroot and holomorphic roots for higher .
