Page 152 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 13: Schwarz's Reflection Principle

Notes
Example: Show that the function defined by

f (z) = 0   t e dt
3
-zt
1
is analytic at all points z for which Re(z) > 0. Find also a function which is analytic continuation
of f (z).
1
3
Solution. f (z) = 0   t e dt
-zt
1
zt

= t 3 e  zt  3t 2 e  2 zt  6t e  zt 3  6 e  4    (Integ. by parts)

  z z  z z 0 
 1  6
= –6 0  4   , if Re(z) > 0

 z  z 4
6
Let f (z) = z 4
2
Then, f (z) = f (z) for Re(z) > 0
2
1
The function f (z) is analytic throughout the complex plane except at z = 0 and f (z) = f (z)  z
2
2
1
s. t. Re(z) > 0. Hence, f (z) is the required analytic continuation of f (z).
2
1
13.3 Schwarz’s Reflection Principle
We observe that some elementary functions f(z) possess the property that for all points z in
some domain. In other words, if w = f(z), then it may happen that i.e. the reflection of z in the real
axis corresponds to one reflection of w in the real axis. For example, the functions
z, z + 1, e , sin z, etc.
2
z
have the above said property, since, when z is replaced by its conjugate, the value of each
function changes to the conjugate of its original value. On the other hand, the functions
iz, z + i, e , (1 + i) sin z etc
2
iz
do not have the said property.
Definition. Let G be a region and G* = {z : z G} then G is called symmetric region if G = G*
If G is a symmetric region then let G = {z  G : I z > 0} G = {z  G : I z < 0} and G = {z  G : I m
+
m
0
m
-
z = 0}.
Theorem (Schwarz’s Reflection Principle). Let G be a region such that G = G* if f : G  G  
0
+
is a 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
+
Proof. For z in G , define g(z) = f(z) and for z in G  G , define g(z) = f(z).
-
+
0
Then g : G   is continuous. We will show that g is analytic. Clearly, g is analytic on G  G -
+
. To show g is analytic on G , let x be a fixed point in G and let R > 0 be such that
0
0
0
B(x ; R)  G.
0
It is sufficient to show that g is analytic on B(x ; R). We shall apply Morera’s theorem.
0
LOVELY PROFESSIONAL UNIVERSITY 145