814hw2sol - Math 814 HW 2 September 29, 2007 p. 43:...

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Unformatted text preview: Math 814 HW 2 September 29, 2007 p. 43: 1,4,6,13,15, p. 54 1, 3 (cos z only). u ( x, y ) = x 3- 3 xy 2 , u ( x, y ) = x/ ( x 2 + y 2 ) , p.43, Exercise 1. Show that the function f ( z ) = | z | 2 = x 2 + y 2 has a derivative only at the origin. On the region U = C- { } we have z = f ( z ) /z . If f ( z ) were analytic at some w U then z , being the product of two functions analytic at w , would itself be analytic at w , which we know is false. Consider now w = 0 . Let > . If | z | < then | f ( z )- f (0) | | z- | = | z | < , so f ( z ) is analytic at . p.43, Exercise 4. Show that (cos z ) =- sin z and (sin z ) = cos z . There are two methods: (cos z ) = 1 2 ( e iz + e- iz ) = i 2 ( e iz- e- iz ) =- sin z, and (cos z ) = X n =0 (- 1) n z 2 n (2 n )! ! = X n =0 (- 1) n 2 n z 2 n- 1 (2 n )! = X n =1 (- 1) n z 2 n- 1 (2 n- 1)! =- sin z. It is similar for (sin z ) . 1 p.43, Exercise 6. Describe the following sets: { z : e z = i } = ( 2 Z + 1 2 ) i { z : e z =- 1 } = (2 Z + 1) i { z : e z =- i } = ( 2 Z- 1 2 ) i { z : cos z = 0 } = ( Z + 1 2 ) { z : sin z = 0 } = Z . p.43, Exercise 13. Let U = C- R . Find all analytic functions f ( z ) on U such that z = ( f ( z )) n ....
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814hw2sol - Math 814 HW 2 September 29, 2007 p. 43:...

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