609S04ex1short

# 609S04ex1short - Math 609 Exam 1 Jerry L Kazdan 1:30 2:50...

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Math 609 Exam 1 Jerry L. Kazdan March 18, 2004 1:30 — 2:50 Directions This exam has two parts, the first is short answer ( 6 points each ) while the second has traditional problems ( 12 points each ). Closed book, no calculators – but you may use one 3” × 5” card with notes. Part A: Short Answer Problems (5 problems, 6 points each) A–1. Find all complex values of 1 i in the form a + ib . A–2. a) For which values of the constant c is u ( x, y ) := 2 y + e 3 y sin cx the real part of an analytic function f = u + iv ? b) For these values of c , find the corresponding function f ( z ). A–3. If a n z n is the power series expansion of 1 cos( z + 1) about z = 0, what is its radius of convergence? A–4. Compute C e z z 2 - 2 z dz , where C is the ellipse x 2 / 25 + y 2 / 9 = 1 (counterclockwise). A–5. Let f ( z ) be holomorphic for 0 < | z | < . If f has no zeroes and | f ( z ) | ≥ | f (2) | in the disk | z - 2 | < 1, what can you conclude about f ( z )? Justify your assertions. Part B: Traditional Problems (6 problems, 12 points each) B–1. Let f ( z ) be holomorphic in {| z

Unformatted text preview: | ≤ 1 } except for a simple pole at z = i/ 2. If f also satisﬁes f ( 1 2 ) = 0 as well as | f ( z ) | ≤ 1 on | z | = 1, show that | f (0) | ≤ 1. B–2. Find a conformal map f ( z ) = u + iv from the unit disk {| z | < 1 } to the ﬁrst quadrant, { u > , v > } . B–3. Give a complete clear proof of the fundamental theorem of algebra: Every nontrivial complex polynomial has at least one root. B–4. Evaluate Z ∞ cos x 1 + x 2 dx . B–5. Let g ( z ) be holomorphic in the closed unit disk D = {| z | ≤ 1 } and assume that | g ( z ) | ≤ 2 for | z | = 1. How many roots does h ( z ) := g ( z ) + 5 z 3-2 have in D ? As usual, justify your assertions. B–6. Let h ( z ) = ∞ X n =1 a n n z , where z = x + iy , and assume the sequence a n is bounded, say | a n | ≤ M . Show that h ( z ) is holomorphic in the half-plane { x > 1 } ....
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