Slit plane - Revision history

Vipul: 4 revisions

2008-05-18T19:18:47Z

4 revisions

← Older revision	Revision as of 19:18, 18 May 2008
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Vipul at 20:13, 27 April 2008

2008-04-27T20:13:46Z

← Older revision		Revision as of 20:13, 27 April 2008
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			{{particular simply connected domain}}
	==Definition==		==Definition==

Vipul at 20:13, 27 April 2008

2008-04-27T20:13:02Z

← Older revision		Revision as of 20:13, 27 April 2008
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	The slit plane also admits a [[holomorphic squareroot]] and holomorphic <math>n^{th}</math> roots for higher <math>n</math>.		The slit plane also admits a [[holomorphic squareroot]] and holomorphic <math>n^{th}</math> roots for higher <math>n</math>.

			More generally, the term '''slit plane''' is used for a [[generalized slit plane]]: the complement in <math>\mathbb{C}</math> of any (not necessarily straight) ray (with endpoint) going off to infinity. Any such slit plane is contractible, though not necessarily star-like, and admits holomorphic logarithms and roots. More generally, any [[simply connected domain]] that is not the whole of <math>\mathbb{C}</math> is contained in a generalized slit plane.

			==Riemann mapping==

			The slit plane admits an easy Riemann mapping -- in fact, it is a very special case of the Riemann mapping theorem. First, choose a holomorphic squareroot on the slit plane, with the property that the squraeroot of a positive real number is is positive squareroot. The image of the slit plane under this mapping is the [[right half-plane]].

			Then the Riemann mapping is given by:

			<math>z \mapsto \frac{\sqrt{z} - 1}{\sqrt{z} + 1}</math>

Vipul at 22:28, 26 April 2008

2008-04-26T22:28:04Z

← Older revision		Revision as of 22:28, 26 April 2008
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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>.

Vipul: New page: ==Definition== The '''slit plane''' is defined as the following open subset of $\mathbb{C}$ : $\mathbb{C} \setminus \{ z \in \R \mid z \le 0 \}$ In other words, it ...

2008-04-26T22:26:49Z

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 ...

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 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:

<math>re^{i\theta} \mapsto \log r + i\theta</math>

where <math>\theta \in (-\pi,\pi)</math> is the principal argument.

Slit plane - Revision history

Vipul: 4 revisions

Vipul at 20:13, 27 April 2008

Vipul at 20:13, 27 April 2008

Vipul at 22:28, 26 April 2008

Vipul: New page: ==Definition== The '''slit plane''' is defined as the following open subset of \mathbb{C}: \mathbb{C} \setminus \{ z \in \R \mid z \le 0 \} In other words, it ...

Vipul: New page: ==Definition== The '''slit plane''' is defined as the following open subset of $\mathbb{C}$ : $\mathbb{C} \setminus \{ z \in \R \mid z \le 0 \}$ In other words, it ...