This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ASSIGNMENT 6 SOLUTIONS MAT 572 A FALL 2007 Problem 1 ( cf. [1, Exercise IV.2 #2]) . Prove that if G C is open and : I G is a rectifiable curve, and : ( I ) G C is continuous and g : G C is defined by g ( z ) = Z ( w, z ) d w, then g is continuous. Also prove that if /z exists and is continuous on ( I ) G , then g is analytic on G , with (1.1) g ( z ) = Z z d w. Proof. For continuity, fix z G and > 0 such that K = B ( z ) G . Then is uniformly continuous on the compact set ( I ) K , so given > 0 we can choose > 0 such that < , and such that  ( w, z ) ( w , z )  < /V ( ) for all ( w, z ) and ( w , z ) in ( I ) K with d (( w, z ) , ( w , z )) < . In particular, for any z G with  z z  < , Proposition III.1.17 gives  g ( z ) g ( z )  = Z ( w, z ) d w Z ( w, z ) d w Z  ( w, z ) ( w, z )  d w  V ( ) Z  d w  = , so g is continuous at z . For analyticity, we only need to establish (1.1), since then continuity of g follows from the above. Again, fix z G and > 0 such that K = B ( z ) G . Also fix > 0, and write 2 for z . Then 2 is uniformly continuous on...
View
Full
Document
 Spring '09

Click to edit the document details