Advanced Calculus fi..

4572 Advanced Calculus, Fifth EditionIt remains to consider the case when f = 0 in a neighborGod of zo. This iscovered by the following theorem.THEOREM 26The zeros of an analytic function are isolated, unless the functionis identically zero; that is, iff (z) is analytic in domain D and f (z) is not identicallyzero, then for each zero zo of f (z) there is a deleted neighborhood of zo in whichf (z) # 0.The proof is given in Section 3.2 of the book by Ahlfors listed at the end of thischapter.The complex number oo has been introduced several times in connection with limitingprocesses, for example, in the discussion of poles in the preceding section. Ineach case, oo has appeared in a natural way as the limiting position of a point recedingindefinitely from the origin. We can incorporate this number into the complexnumber system with special algebraic rules:Expressions such as oo + oo, oo - oo, and oo/oo are not defined.A function f (z) is said to be analytic in a deleted neighborhood of oo if f (z) isanalytic for lzl > R1 for some R1. In this case the Laurent expansion with R2 = ooand zo = 0 is available, and we haveIf there are no positive powers of z here, f (z) is said to have a removable singularityat oo, and we make f analytic at oo by defining f (oo) = ao:a-1 a-nf(z)=ao+-+...+-+... ( 1 ~ 1 > RI). (8.68)z znThis is clearly equivalent to the statement that if we set zl = l/z, then f (z) becomesa function of zl with removable singularity at zl = 0.If a finite number of positive powers occurs, we have, with N 2 1,where @(z) is analytic at oo and @(GO) = a~ # 0. In this case, f (z) is said to havea pole of order N at oo. The same holds for f (1 /zl ) at z = 0. Furthermore,lim f(z) = ao.z+ a0(8.70)