solutions and samples

solutions and samples - b u is harmonic on D it follows...

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Math 380 Bonus problem Due 4/14 Worth 2 homework points. 1. Let D denote the open unit disk and let @D denote its boundary, the unit circle. Suppose u ( e it ) is a continuous real-valued function on @D . For re in D , de°ne b u ° re ± = 1 ° r 2 2 ° Z 2 ± 0 u ( e it ) 1 ° 2 r cos( ± ° t ) + r 2 dt: This is the Poisson integral of u , which de°nes a harmonic extension of u into the unit disk. The development of this integral was outlined in class. This assignment is to °ll in some of the gaps. 2. For °xed z 0 = re 2 D; de°ne g ( w ) = z 0 ° w 1 ° z 0 w : The function g provides a one-to-one analytic mapping of D [ @D onto D [ @D . (a) Use the polar coordinate formulation of Laplace±s equation to verify that b u is harmonic in D: (b) If z = g ( w ) , solve for w in terms of z and verify that w = z 0 ° z 1 ° z 0 z : (c) De°ne ( w ) = b u ( g ( w )) : Since g : D ! D is analytic and
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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...
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