Applied complex variables axam

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FINAL EXAM (VERSION C)

This test has 8 pages and 7 problems.

No calculators are allowed. You are permitted only a pencil or pen. Work out everything as far as you can before making decimal approx-imations.

1. Find all solutions z to the equation tanhz = 1 .

Date_{: May 6, 2002.}

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3. Calculate the integral Z

C

cosz z(z2_{+ 8)}dz

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4. Calculate the integral Z

|z|=1

cos (1/z)dz z2_{sin (1/z)}.

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5. Show that the map

w=

z−₁

z+ 1 1/2

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–1 –0.5

0.5 1

–1 –0.5 0.5 1 1.5 2

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7. Suppose that f(z) is an entire analytic function, and that the real part off(z), which we will callu(x, y), has an upper bound:

u(x, y)≤_u₀