# 06 - ASSIGNMENT 6 Â SOLUTIONS MAT 572 A Â FALL 2007 Problem...

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Unformatted text preview: ASSIGNMENT 6 Â· SOLUTIONS MAT 572 A Â· FALL 2007 Problem 1 ( cf. [1, Exercise IV.2 #2]) . Prove that if G âŠ† C is open and Î³ : I â†’ G is a rectifiable curve, and Ï• : Î³ ( I ) Ã— G â†’ C is continuous and g : G â†’ C is defined by g ( z ) = Z Î³ Ï• ( w, z ) d w, then g is continuous. Also prove that if âˆ‚Ï•/âˆ‚z exists and is continuous on Î³ ( I ) Ã— G , then g is analytic on G , with (1.1) g ( z ) = Z Î³ âˆ‚Ï• âˆ‚z d w. Proof. For continuity, fix z âˆˆ G and Î· > 0 such that K = B Î· ( z ) âŠ† G . Then Ï• is uniformly continuous on the compact set Î³ ( I ) Ã— K , so given > 0 we can choose Î´ > 0 such that Î´ < Î· , and such that | Ï• ( w, z )- Ï• ( w , z ) | < /V ( Î³ ) for all ( w, z ) and ( w , z ) in Î³ ( I ) Ã— K with d (( w, z ) , ( w , z )) < Î´ . In particular, for any z âˆˆ G with | z- z | < Î´ , Proposition III.1.17 gives | g ( z )- g ( z ) | = Z Î³ Ï• ( w, z ) d w- Z Î³ Ï• ( w, z ) d w â‰¤ Z Î³ | Ï• ( w, z )- Ï• ( w, z ) || d w | â‰¤ V ( Î³ ) Z Î³ | 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 âˆˆ G and Î· > 0 such that K = B Î· ( z ) âŠ† G . Also fix > 0, and write Ï• 2 for âˆ‚Ï• âˆ‚z . Then Ï• 2 is uniformly continuous on...
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06 - ASSIGNMENT 6 Â SOLUTIONS MAT 572 A Â FALL 2007 Problem...

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