Page 150 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 13: Schwarz's Reflection Principle

Remark. The circles C and C touch internally at z = –1 which is a singularity for both f (z) and Notes
2
1
1
f (z) i.e. z = –1 is a singularity of the complete analytic function whose two representations
2
(members) are f (z) & f (z).
1
2
Example: Show that the functions defined by the series
1 + az + a z + ..
2
2
1 (1 a)z (1 a) z 2
2


and  
1 z (1 z) 2 (1 z) 3



are analytic continuations of each other.
1
Solution. The first power series represents the function f (z) = 1 az and has the circle of
1

1 1
convergence C given by |az| = 1 i.e. |z| = |a| . The only singularity is at the point z = (a  0)
a
1
on the boundary of the circle. The second series has the sum function
1 (1 a)z (1 a) z 2
2


f (z) = 1 z  (1 z) 2  (1 z) 3 .....
2





2
= 1   1    1 a   z    1 a   2 z .........  



1 z    1 z   1 z   

= 1 1   1 a   z  1
1 z  1 a   1 z 



1    z

 1 z 
1 1 z 1

= . 
1 z 1 az 1 az



and has the circle of convergence C given by
2
z(1 a) 1 i.e. |z(1 – a)| = |1 – z|

1 z 

i.e. |z(1-a)| = |1-z| 2
2
i.e. z z (1-a) = (1-z) , where a is assumed to be real and a > 0
2
i.e. z z (1-a) = 1 -(z + z ) + z z
2
i.e. (x + y ) (1+a - 2a) = 1 -2x + x + y , z = x + iy
2
2
2
2
2
i.e. (x + y ) a (a-2) = 1 -2x
2
2
2x 1
i.e. x + y -   0
2
2
a(2 a) a(2 a)


 1  2  1 a  2

2
i.e.  x    (y 0)   



 a(2 a)   a(2 a) 
LOVELY PROFESSIONAL UNIVERSITY 143