Page 154 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 13: Schwarz's Reflection Principle

Notes



  f| = |  f   f   f   f| |  f   f||  f| |  f
T [a,b] [ , ] [ ,a] [b, ]  [a,b] [ , ] [ ,a] [b, ]







 l + 4M

Choosing  > 0 s. t.  < . Then


| f| < (l + 4M). Since  is arbitrary it follows that f  0. Hence, f must be analytic.
T T
13.4 Summary
An analytic function f(z) with its domain of definition D is called a function element and

is denoted by (f, D). If zD, then (f, D) is called a function element of z. Using this notation,
we may say that (f , D ) and (f , D ) are direct analytic continuations of each other iff
1
2
2
1
D  D   and f (z) = f (z) for all zD  D .
1
1
2
1
2
2
Remark. We use the word ‘direct’ because later on we shall deal with continuation along
a curve. i.e. just to distinguish between the two.
Analytic continuation along a chain of Domain. Suppose we have a chain of function
elements (f , D ), (f , D ), , (f , D ), ,(f , D ) such that D and D have the part D in
1
k
2
K
n
2
12
2
1
n
1
common, D and D have the part D is common and so on. If f (z)= f (z) in D , f (z) = f (z)
2
1
3
2
23
12
3
2
in D and so on, then we say that (f , D ) is direct analytic continuation of (f , D ). In this
23
K
K-1
k
K-1
way, (f , D ) is analytic continuation of (f , D ) along a chain of domains D , D , , D .
1
1
2
n
n
1
n
Without loss of generality, we may take these domains as open circular discs. Since (f ,
K-1
D ) and (f , D ) are direct analytic continuations of each other, thus we have defined an
K1
K
K
equivalence relation and the equivalence classes are called global analytic functions.
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
2
n
n
1
1
2
continuations 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
0
0
singularity of f(z). Obviously, the singularity of f(z) is also a singularity of the corresponding
complete analytic function F(z).
13.5 Keywords
Analytic function: An analytic function f(z) with its domain of definition D is called a function
element and is denoted by (f, D). If zD, then (f, D) is called a function element of z.
Analytic continuations: Using this notation, we may say that (f , D ) and (f , D ) are direct
1
1
2
2
analytic continuations of each other iff D  D   and f (z) = f (z) for all zD  D .
1
2
1
2
1
2
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