Page 155 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Uniqueness of Direct Analytic Continuation: There cannot be two different direct analytic
continuations of a function.
Uniqueness of Analytic Continuation along a Curve: Analytic continuation of a given function
element along a given curve is unique. In other words, if (f , D ) and (g , E ) are two analytic
n
n
m
m
continuations of (f , D ) along the same curve g defined by
1
1
z = z(t) = x(t) + iy(t), a  t  b.
Then f = g on D  E m
n
n
m
Schwarz’s Reflection Principle: Let G be a region such that G = G* if f : G  G   is a
0
+
continuous function which is analytic on G and f(x) is real for x in G then there is an analytic
0
+
function g : G   s.t. g(z) = f(z) for all z in G G .
0
+
13.6 Self Assessment
1. An ................. 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.
2. Using this notation, we may say that (f , D ) and (f , D ) are direct ................. of each other
2
2
1
1
iff D  D   and f (z) = f (z) for all zD  D .
1
1
2
2
1
2
3. There cannot be two ................. analytic continuations of a function.
4. Theorem (Schwarz’s Reflection Principle). Let G be a region such that G = G* if f : G 
+
G   is a continuous function which is analytic on G and f(x) is real for x in G then there
0
+
0
is an analytic function g : ................. for all z in G G .
+
0
13.7 Review Questions
1. Given the identity sin z + cos z = 1 holds for real values of z, prove that it also holds for all
2
2
complex values of z.
2. Explain how it is possible to continue analytically the function
f(z) = 1 + z + z + + z +
n
2
outside the circle of convergence of the power series.
3. Show that the function
1 z z 2
f(z) =    ...
a a 2 a 3
can be continued analytically.
4. Show that the circle of convergence of the power series
f(z) = 1 + z + z + z + z +
2
4
8
is a natural boundary of its sum function.
5. Discuss the concept of complex analytic functions.
6. Describe the Schwarz's reflection principle.

148 LOVELY PROFESSIONAL UNIVERSITY