M461practicesol2

M461practicesol2 - MAT 461 - Test 2 Practice Posted...

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Unformatted text preview: MAT 461 - Test 2 Practice Posted November 13 1. (a) Let f ( z ) = 1 z ( z- 1) . Give the two Laurent Series centered at z = 0, one valid < | z | < 1, one valid 1 < | z | < . (b) Give the Laurent series centered at z = 2 for f ( z ) = sin z ( z- 2 ) 3 . What is the residue of f ( z ) and z = 2 ? 2. Let f ( z ) be entire. Assume that | f ( z ) | > m for all complex numbers z , where m is a positive constant. Prove that f ( z ) is constant. Let C be the circle | z | = 2 , positively oriented. 3. Compute the integral integraldisplay C 1 z 2 ( z 3 i ) dz 4. Compute the integral integraldisplay C ( z 1) 3 e 1 z- 1 dz 5. Compute the integral integraldisplay C e 1 z cos( 1 z ) dz 6. Define log z with the branch cut at < < . Compute the integral integraldisplay C sin(log( z + 3)) ( z 4 i ) 2 dz Solutions 1. (a) 1 z ( z 1) = 1 z 1 1 z = 1 z summationdisplay n =0 z n this sum works for | z | < 1 = 1 z 1 z z 2 =...
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M461practicesol2 - MAT 461 - Test 2 Practice Posted...

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