T1sol - Complex Variables MAT334 T ERM TEST 1 Solution...

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Unformatted text preview: Complex Variables MAT334 T ERM TEST 1 Solution February 11, 2004. 1. Let A be a subset of C . State precise definitions of when is z an interior point of A and what it means for A to be open. Ans: z is an onterior point of A if there exists r > such that B ( z, r ) A . The set A is open if all of its points are interior points. 2. Let f ( z ) = 2 i z +1 . Show that the image of the disc B (0 , 1) , by function f , is contained in the set { w | Im w > 1 } . Ans: Let z B (0 , 1) . That is let | z | < 1 . Then z z = | z | 2 < 1 . It suffices to show that Im 2 i z +1 > 1 . Im 2 i z + 1 = Re 2 z + 1 = Re 2( z + 1) z z + z + z + 1 = 2 Re z + 2 z z + 2 Re z + 1 > 1 . To establish the last inequality we have used that z z < 1 and hence 2 Re z + z z + 1 < 2 Re z + 2 . 3. Let v ( x, y ) = e 2 y (cos 2 x- sin 2 x ) . Show that v is a harmonic function. Then find a harmonic conjugate of v . Ans: Taking the partial derivatives yields v xx + v yy =- 4 e 2 y (cos 2 x- sin 2 x ) + 4 e 2 y (cos 2...
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T1sol - Complex Variables MAT334 T ERM TEST 1 Solution...

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