MA530_JAN95 - (0) = 0, ( 1 2 ) = 1. 1 5. a) How many roots...

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QUALIFYING EXAMINATION JANUARY 1995 MATH 530 1. Let f ( z )= a 1 z + a 2 z 2 + a 3 z 3 + ... be an analytic function at 0 and a 2 6 =0 . Express the residue of 1 /f 2 at 0 in terms of a i . Remark : Don’t forget the case a 1 =0. 2. Find an analytic function f such that | f ( x + iy ) | = e xy . 3. Find all complex solutions of the equation cos z =2. 4. Find the conformal mapping ϕ
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Unformatted text preview: (0) = 0, ( 1 2 ) = 1. 1 5. a) How many roots does this equation z 4 + z + 5 = 0 have in the rst quadrant. b) How many of them have argument between 4 and 2 ? 6. Compute Z | z | =1 e z z-n dz, where n is an integer. 7. Show that an isolated singularity of f cannot be a pole of sin f . 2...
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MA530_JAN95 - (0) = 0, ( 1 2 ) = 1. 1 5. a) How many roots...

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