Unformatted text preview: b u is harmonic on D , it follows that ’ is harmonic on D: Show that b u ( z ) = ’ (0) : (d) Since ’ is harmonic, it satis&es the Mean Value Property, so b u ( z ) = ’ (0) = 1 2 & Z 2 ± ’ & e is ± ds = 1 2 & Z 2 ± b u ( z & e is 1 & z e is ) ds = 1 2 & Z 2 ± u ( z & e is 1 & z e is ) ds: The quantity z & e is 1 & z e is has modulus 1 so can be expressed in the form e it : Setting e it = z & e is 1 & z e is , apply the change of variables formula to verify that b u ( z ) = 1 & r 2 2 & Z 2 ± u ( e it ) 1 & 2 r cos( ± & t ) + r 2 dt: 1...
View
Full Document
 Spring '11
 Staff
 Unit Circle, Laplace, Unit disk, @D

Click to edit the document details