https://companal.subwiki.org/w/index.php?action=history&feed=atom&title=StripStrip - Revision history2026-06-16T18:16:14ZRevision history for this page on the wikiMediaWiki 1.41.2https://companal.subwiki.org/w/index.php?title=Strip&diff=505&oldid=prevVipul: 1 revision2008-05-18T19:19:05Z<p>1 revision</p>
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</td></tr></table>Vipulhttps://companal.subwiki.org/w/index.php?title=Strip&diff=504&oldid=prevVipul: New page: {{generic simply connected domain}} ==Definition== A '''strip''' in <math>\mathbb{C}</math> is an open connected subset in <math>\mathbb{C}</math> bounded by two parallel lines. Some spe...2008-04-27T20:51:17Z<p>New page: {{generic simply connected domain}} ==Definition== A '''strip''' in <math>\mathbb{C}</math> is an open connected subset in <math>\mathbb{C}</math> bounded by two parallel lines. Some spe...</p>
<p><b>New page</b></p><div>{{generic simply connected domain}}<br />
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==Definition==<br />
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A '''strip''' in <math>\mathbb{C}</math> is an open connected subset in <math>\mathbb{C}</math> bounded by two parallel lines. Some special kinds of strips:<br />
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* '''Vertical strip''': A strip bounded by vertical lines. It is defined by saying that the [[real part]] must lie within an open interval<br />
* '''Horizontal strip''': A strip bounded by horizontal lines. It is defined by saying that the imaginary part must lie within an open interval<br />
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==Riemann mapping==<br />
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Any two strips are equivalent why translations and multiplications by complex numbers, so it suffices to give a [[Riemann mapping]] to the [[open unit disk]], for one particular strip. consider the strip bound by the horizontal lines <math>y = 0</math> and <math>y = \pi i</math>. The Riemann mapping is given by:<br />
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<math>z \mapsto \frac{e^z - i}{e^z + i}</math><br />
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The [[complex exponential]] maps the strip to the [[upper half-plane]], while the fractional linear transformation maps the upper half-plane to the [[open unit disk]].<br />
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==Relation with other domains==<br />
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* [[Slit plane]]: The strip bounded by the horizontal lines <math>y = \pm \pi i</math> gets mapped to the slit plane under the [[complex exponential]].<br />
* [[Upper half-plane]]: The strip bounded by the horizontal lines <math>y = 0</math> and <math>y = \pi i</math> gets mapped to the upper half-plane under the complex exponential.</div>Vipul