7.2 • TAYLOR SERIES REPRESENTATIONS 259A singular point of a function is a point at which the function fails to be
analytic. You will see in Section 7.4 that singular points of a function can be
classified according to how badly the function behaves at those points. Loosely
speaking, a nonremovable singular point of a function has the property that it
is impossible to redefine the value of the function at that point so as to make
it analytic there. For example, the function f (z) = 1 ~,. has a nonremovable
singularity at z = 1. We give a formal definition of this concept in Section 7.4,
but with this language we can nuance Taylor's theorem a bit.
t Corollary 7 .3 Suppose that f is analytic in the domain G that contains the
point °'· Let zo be a nonremovable singular point of minimum distance to the
point °'· If lzo -al = R, then(^00) J <kl(~\ k
i. the Taylor series I: ~ (z - a) converges to f (z) on all of DR (0<), and
k=O
ii. if lz 1 - 0:1 = S > R, the Taylor series f / <~,<al (z 1 - a)k does not converge
k=O
to f (z1).
P roof Taylor's theorem gives us part (i) immediately. To establish part (ii),
we note that if lzo -0: 1 = R, then zo E D s (a) whenever S > R. If for some z1 ,
with lz1 -al = S > R, the Taylor series converged to f (z1), then according to
00 (k)
Theorem 4.17, the radius of convergence of the series I; / k!(a) (z - a)" would
k=O