Infinity Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ====Complex analysis==== [[File:Riemann sphere1.svg|thumb|right|250px|By [[stereographic projection]], the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the [[Riemann sphere]].]] In [[complex analysis]] the symbol <math>\infty</math>, called "infinity", denotes an unsigned infinite [[Limit (mathematics)|limit]]. The expression <math>x \rightarrow \infty</math> means that the magnitude <math>|x|</math> of ''<math>x</math>'' grows beyond any assigned value. A [[point at infinity|point labeled <math>\infty</math>]] can be added to the complex plane as a [[topological space]] giving the [[one-point compactification]] of the complex plane. When this is done, the resulting space is a one-dimensional [[complex manifold]], or [[Riemann surface]], called the extended complex plane or the [[Riemann sphere]].<ref>{{Cite book|title=Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory|first1=Murali|last1=Rao|first2=Henrik|last2=Stetkær|publisher=World Scientific|year=1991|isbn=9789810203757|page=113|url=https://books.google.com/books?id=wdTntZ_N0tYC&pg=PA113}}</ref> Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables [[division by zero]], namely <math>z/0 = \infty</math> for any nonzero [[complex number]] ''<math>z</math>''. In this context, it is often useful to consider [[meromorphic function]]s as maps into the Riemann sphere taking the value of <math>\infty</math> at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of [[Möbius transformation]]s (see [[Möbius transformation#Overview|Möbius transformation § Overview]]). Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page