Advanced Calculus fi..

Chapter 8 Functions of a Complex Variable 569neighborhood of zo, that is, in the circular domain lz - zol < R2 minus its centerzo. If R2 = oo, we can say similarly that the series represents f (z) in a deletedneighborhood of z = oo.8.13 ISOLATED SINGULARITIES OF AN ANALYTIC ,FUNCTION ZEROS AND POLESLet f (z) be defined and analytic in domain D. We say that f (z) has an isolatedsingularity at the point zo iff (z) is analytic throughout a neighborhood of zo exceptat zo itself; that is, to use the term mentioned at the end of the preceding section,f (z) is analytic in a deleted neighborhood of zo but not at zo. The point zo is then aboundary point of D and would be called an isolated boundarypoint (see Fig. 8.14).A deleted neighborhood 0 < lz - zol < R2 forms a special case of the annulardomain for which Laurent's theorem is applicable. Hence in this deleted neighborhood,f (z) has a representation as a Laurent series:The form of this series leads to a classification of isolated singularities into threefundamental types:Case I. No terms in negative powers of z - zo appeal: In this case the series isa Taylor series and represents a function analytic in a neighborhood of zo. Thusthe singularity can be removed by setting f (zo) = ao. We call this a removablesingularity of f (2). It is illustrated byz2 z4I--+--...--sin z-z 3! 5!at z = 0. In practice we automatically remove the singularity by defining the functionproperly.1Figure 8.14Isolated singularity.