1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 529

= J [l

00

f((, t) dt] d( :S €L(C)

c

Hence

j [fo

00

t((,t)dt] d( = J~foT [iat((,t)d(] dt

c

and this is equivalent to (8.2-3).

8.3 The Cauchy-Taylor Expansion Theorem

We have seen in Section 8.1 (Corollary 8.4) that power series with nonzero
radius of convergence represent functions which are analytic inside the cir-
cle of convergence. In this section we prove that, conversely, an analytic
function in a certain region can be expanded at each point a of the region
in a power series of the form I:~=O an ( Z - a r which represents the function
in some open disk contained in that region. In fact, we shall prove that
the Taylor expansion established in the real domain for functions having
infinitely many derivatives is still valid in the complex case with some ad-
vantage, since in the last case the so-called remainder of the series always
tends to zero as n -+ oo.

Theorem 8.5 (Cauchy-Taylor Expansion Theorem). Let f be a (single-

valued) i;i,nalytic function in a region D and a a point in D. Then f(z) can
be expressed as a series in powers of z - a with a radius of convergence

extending from a to the nearest (nonremovable) singularity of f. This

expansion is unique and given by the formula

f(z) = f(a) + f'(a) (z - a)+···+ f(n)(a) (z -at+···

1! n!

~ J(k)(a) k

= L.J-k-

1

-(z-a)

. k=O

(8.3-1)

In the case of an analytic branch of a multiple-valued function the region
of validity of (8.3-1), i.e., the region in which the series converges to f(z),
could be a portion of the disk of convergence of the series (see Example
7 below).

Proof Let is-al = R, wheres is the singularity off nearest a. If there are
several singularities at the same distance from a, let s be anyone of them,

and if f is an entire function, put R = oo. For R < oo, consider the circle

C: ( - a = Reit, 0 :S t :S 27r, and let z be any point inside C, i.e., such

• I

I I