Mathematics of Physics and Engineering

Convergence 211

where

By (4.3.6), the radius of convergence of the power series in (4.3.8) is at least
p. This shows that an analytic function can be represented by a convergent
power series.
Step 2. Let us show that a convergent power series defines a continuous
function. Define g(z) = J2k>oak(z ~ zo)k, \z — ZQ\ < R. We will show
that the function g is continuous for \z — ZQ\ < R, that is, for every z\ in
the disk of convergence and for every e > 0, there exits a S > 0 so that
\g(zi) — g(z)\ < E as long as \z\ — z\ < S and \z — ZQ\ < R; the interested
reader can then use similar arguments to prove that g is analytic. To prove
the continuity of g, fix a z\ with \z\ — ZQ\ < R and an e > 0. Next, let
us define r = (R+\z\ — zo)/2 and find TV so that J2k>N+i \ak\rk < e/4-
Such an N exists because r < R and the power series converges absolutely
inside the disk of convergence. Now consider gN(z) = J2k=i akZk- Being a
polynomial, gN(z) is a continuous function at z\, and therefore there exists
a <5i > 0 so that \gw(z) — g(z)\ < e/2 as long as \z — z\ < 6. Finally,
define S as the minimum of Si and (R — \z\ — ZQ)/A. By construction, if
l^i — z\ < S, then \ZQ — z\ < r (draw a picture). Now let us look at the
value of \g{zi) — g(z)\ for \z — z\ < 6. By the triangle inequality,

\g(zi)-g(z)\<\gN(z 1 )-gN(z)+ ^ \ak\ \zi-zQ\k+ ]T \ak\ \z-zQ\k;

k>N+l k>N+l

(4.3.10)

by the choice of N and J, |gjv(zi) — 9JV(Z)| < e/2, and, by the choice of r,

X) \ak\\zi-z 0 \k+ ]T \ak\\z-zQ\k <2 J2 \ak\rk<e/2.

k>N+l k>N+l k>N+l

As a result, \g(zi) — g(z)\ < e, which proves the continuity of g.

The following exercise completes the proof of the theorem. •

EXERCISE 4.3.8/^1 (a) Use the same arguments as in Step 2 above to show

that g(z) = J2k>o °fe(z ~ z°)k ls differentiable for \z — z 0 | < R and g'{z) =

Sfc>o kak{z—zo)k~l • Thus, g is indeed an analytic function for \z—ZQ\ < R.

Hints: (i) you can differentiate the polynomials; (ii) by Exercise 4-3.6, the power

series ^2k>0 fcafcZ*-1 has the radius of convergence equal to R. (b) Use the same

arguments to justify the switch of summation and integration in (4-3.7).

Hints: (i) you can do this switching when the number of terms is finite, (ii)