section_path: Mathematics/Calculus/Complex analysis
section_name: Complex analysis
paragraph_text: Complex analysis
  
  In complex analysis the symbol ∞ {\displaystyle \infty }, called "infinity", denotes an unsigned infinite limit. The expression x → ∞ {\displaystyle x\rightarrow \infty } means that the magnitude | x | {\displaystyle |x|} of x {\displaystyle x} grows beyond any assigned value. A point labeled ∞ {\displaystyle \infty } 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. 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 z / 0 = ∞ {\displaystyle z/0=\infty } for any nonzero complex number z {\displaystyle z}. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of ∞ {\displaystyle \infty } 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 transformations (see Möbius transformation § Overview).