Page 146 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 13: Schwarz's Reflection Principle

The first thing to observe here is that, when the continuation has been carried out, there must be Notes
at least one singularity of the complete analytic function on the circle of convergence C . For if
0
there were not, we would construct, by analytic continuation, an analytic function which is
equal to f(z) within C but is regular in a larger concentric circle C . The expansion of this
0
0
function as a Taylor’s series in powers of z -z would then converge everywhere within C ,
0
0
which is, however, impossible since the series would necessarily be the original series, whose
circle of convergence is C .
0
To carry out the analytic continuation, we take any fixed point z within C , and calculate the
1
0
values of f(z) and its successive derivatives at that point from the given power series by repeated
term-by-term differentiation. We then form the Taylor’s series
n
 f (z )
 1 (z - z ) n (1)
1
0 | n
whose circle of convergence is C , say. Let denote the circle with centre z which touches C 0
1
1
1
internally. By Taylor’s theorem, this new power series is certainly convergent within g and has
1
the sum f(z) there. Hence the radius of C cannot be less than that of g . There are now three
1
1
possibilities :
(i) C may have a larger radius than  . In this case C lies partially outside C and the new
1
0
1
1
power series (1) provides an analytic continuation of f(z). We can then take a point z 2
within C and outside C and repeat the process as far as possible.
0
1
(ii) C may be a natural boundary of f(z). In this case, we cannot continue f(z) outside C and
0
0
the circle C touches C internally, no matter what point z within C was chosen.
0
1
0
1
(iii) C may touch C internally even if C is not a natural boundary of f(z). The point of contact
0
0
1
of C and C is then a singularity of the complete analytic function obtained by the analytic
0
1
continuation of the sum function of the original power series, since there is necessarily
one singularity of the complete analytic function on C and this cannot be within C .
0
1

Thus, if C is not a natural boundary for the function f(z) =  a (z-z ) , this process of forming
n
n
0
0
n 0

the new power series of the type (1) provides a simple method for the analytic continuation of
f(z), know as power series method.
Remark. Power series method is also called standard method of analytic continuation.
Example: Explain how it is possible to continue analytically the function
f(z) = 1 + z + z + + z +
2
n
outside the circle of convergence of the power series.
Solution. The circle of convergence of the given power series is |z| = 1. Denoting it by C , we
0
observe that within C the sum function f(z) = (1-z) is regular. Further, this function is regular
-1
0
in any domain which does not contain the point z = 1. We carry out the analytic continuation by
means of power series. If a is any point inside C such that a is not real and positive, then
0
|1 – a| > 1 – |a| (2)
Now, the Taylor’s expansion of f(z) about the point z = a is given by
n
 f (a)
 (z - a) n (3)
n 0 | n

LOVELY PROFESSIONAL UNIVERSITY 139