Page 144 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 13: Schwarz's Reflection Principle

13.2 Complete Analytic Function Notes

Suppose that f(z) is analytic in a domain D. Let us form all possible analytic continuations of
(f, D) and then all possible analytic continuations (f , D ), (f , D ), , (f , D ) of these continuations
2
1
1
2
n
n
such that

 f (z) if z D 1
1
 f (z) if z D

 2 2
  ..........................
F(z) = 
 ..........................
 ...........................


  f (z) if z D n
n
Such a function F(z) is called complete analytic function. In this process of continuation, we may
arrive at a closed curve beyond which it is not possible to take analytic continuation. Such a
closed curve is known as the natural boundary of the complete analytic function. A point lying
outside the natural boundary is known as the singularity of the complete analytic function. If no
analytic continuation of f(z) is possible to a point z , then z is a singularity of f(z). Obviously, the
0
0
singularity of f(z) is also a singularity of the corresponding complete analytic function F(z).
Theorem (Uniqueness of Direct Analytic Continuation)
There cannot be two different direct analytic continuations of a function.
Proof. Let f (z) be an analytic function regular in the domain D and let f (z) and g (z) be two
2
1
1
2
direct analytic continuations of f (z) from D into the domain D via D which is the domain
1
2
12
1
common to both D and D . Then by definition of analytic continuation, f (z) and g (z) are two
1
2
2
2
functions analytic in D such that
2
f (z) = f (z) and f (z) = g (z)
2
1
2
1
at all points z in D i.e. f (z) = g (z) in D . Thus f (z) and g (z) are two functions analytic in the
2
12
12
2
2
2
domain D such that they coincide in a part D of D . It follows from the well known result that
12
2
2
they coincide throughout D . i.e. f (z) = g (z) throughout D . Hence, the result.
2
2
2
2
Example: Given the identity sin z + cos z = 1 holds for real values of z, prove that it also
2
2
holds for all complex values of z.
Solution. Let f(z) = sin z + cos z-1 and let D be a region of the z-plane containing a portion of
2
2
x-axis (real axis). Since sin z and cos z are analytic in D so f(z) is also analytic in D. Also f(z) = 0
on the x-axis. Hence, by the well known result, it follows that f(z) = 0 identically in D, which
shows that sin z + cos z = 1 for all z in D. Since D is arbitrary, the result holds for all values of
2
2
z.
Remark. This method is useful in proving for complex values many of the results true for real
values.
Analytic continuation along a curve
Let  be a curve in the complex pane having equation
z = z(t) = x(t) + iy(t), a  t  b.
We take the path along  to be continuous. Let a = t  t   t = b be the portion of the interval.
0
1
n
If there is a chain (f , D ), (f , D ), ,(f , D ) of function elements such that (f K+1 , D K+1 ) is a direct
n
1
2
1
2
n
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