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MA530_JAN95

# MA530_JAN95 - ϕ(0 = 0 ϕ ± 1 2 = ± 1 1 5 a How many...

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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 of the following domain onto the unit disk with

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Unformatted text preview: ϕ (0) = 0, ϕ ( ± 1 2 ) = ± 1. 1 5. a) How many roots does this equation z 4 + z + 5 = 0 have in the ﬁrst quadrant. b) How many of them have argument between π 4 and π 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 - ϕ(0 = 0 ϕ ± 1 2 = ± 1 1 5 a How many...

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