Exam_in_CA_-_2 - such that the transformation 2 1 1 ) (...

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Complex Variables and Applications (Total score: 70 ) Class: Name: Number: 1 2 3 4 5 6 7 8 total score 1. ( ' 5 4 × ) Find the values of the following integrals: (1) ( 29 ( 29 = + + 3 3 4 2 2 15 , 2 1 z dz z z z (2) = - + 1 2 2 1 sin z az dz e az z z (3) ( 29 ( 29 2 2 sin ; 1 2 x dx x x +∞ -∞ + + (4) + π 2 0 2 ; cos 2 sin dx x x 2. ( 29 ' 8 Find the function ) , ( ) , ( ) ( y x iv y x u z f + = analytic that satisfies the condition 3 2 2 3 2 6 3 ) , ( 2 ) , ( y y x xy x y x v y x u - + - = + . 3. ( 29 ' 8 Assume that ( 29 - = C a d z e z f , ) ( 2 ξ where C is the unit circle . , θ - = i e z Find the values of 2 i f and 2 ' ' i f . 4. ( 29 ' 4 2 × For every 1,2,3, , n = L prove the following results: (1) ( 29 n r r n d n r e ! 2 sin cos 2 0 cos φ = - (2) ( 29 0 sin sin 2 0 cos = - d n r e r 1
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5. ( 29 ' 6 Write the all possible series expansion in powers of 1 + z that represent the function ( 29 . 1 1 ) ( 2 - + = z z z z f in certain domains, and specify these domains. 6. ( 29 ' 6 Determine a conformal mapping ) ( z f from { } 2 3 1 + < = z and z z D to { } . 0 Re 1 < = z z z 7. ( 29 ' 6 Find the domain
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Unformatted text preview: such that the transformation 2 1 1 ) ( -+-= = z z i z f w maps the domain { } 2 , 2 &lt;-+ = i z i z z D onto . 8. ( 29 ' 8 (1) Suppose that the function ) ( z f is analytic on the domain . 1 &lt;-&lt; z z Prove that z is a pole of the function ) ( z f if and only if . ) ( lim = z f z z (2) Suppose that is a bounded domain, and suppose that ) ( z f and ) ( z g are analytic on . Prove that if ) ( ) ( z g z f = for x , then ) ( ) ( z g z f for . x 2...
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Exam_in_CA_-_2 - such that the transformation 2 1 1 ) (...

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